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mathematical sciences publishers

Volume 1 No 3 2008

THE PSEUDOSPECTRUM OF SYSTEMS OF SEMICLASSICAL OPERATORS

NILS DENCKER

ANALYSIS AND PDEVol 1 No 3 2008

NILS DENCKER

The pseudospectrum (or spectral instability) of non-self-adjoint operators is a topic of current interest inapplied mathematics In fact for non-self-adjoint operators the resolvent could be very large outside thespectrum making numerical computation of the complex eigenvalues very hard This has importancefor example in quantum mechanics random matrix theory and fluid dynamics

The occurrence of false eigenvalues (or pseudospectrum) of non-self-adjoint semiclassical differentialoperators is due to the existence of quasimodes that is approximate local solutions to the eigenvalueproblem For scalar operators the quasimodes appear generically since the bracket condition on theprincipal symbol is not satisfied for topological reasons

In this paper we shall investigate how these results can be generalized to square systems of semiclas-sical differential operators of principal type These are the systems whose principal symbol vanishes offirst order on its kernel We show that the resolvent blows up as in the scalar case except in a nowheredense set of degenerate values We also define quasisymmetrizable systems and systems of subelliptictype for which we prove estimates on the resolvent

1 Introduction

In this paper we shall study the pseudospectrum or spectral instability of square non-self-adjoint semi-classical systems of principal type Spectral instability of non-self-adjoint operators is currently a topicof interest in applied mathematics see [Davies 2002] and [Trefethen and Embree 2005] It arises fromthe fact that for non-self-adjoint operators the resolvent could be very large in an open set containing thespectrum For semiclassical differential operators this is due to the bracket condition and is connectedto the problem of solvability In applications where one needs to compute the spectrum the spectralinstability has the consequence that discretization and round-off errors give false spectral values so-called pseudospectra see [Trefethen and Embree 2005] and references there

We shall consider bounded systems P(h) of semiclassical operators given by (22) and we shallgeneralize the results of the scalar case in [Dencker et al 2004] Actually the study of unboundedoperators can in many cases be reduced to the bounded case see Proposition 220 and Remark 221We shall also study semiclassical operators with analytic symbols in the case when the symbols can beextended analytically to a tubular neighborhood of the phase space satisfying (23) The operators westudy will be of principal type which means that the principal symbol vanishes of first order on thekernel see Definition 31

The definition of semiclassical pseudospectrum in [Dencker et al 2004] is essentially the bracketcondition which is suitable for symbols of principal type By instead using the definition of (injectivity)

MSC2000 primary 35S05 secondary 58J40 47G30 35P05Keywords pseudospectrum spectral instability system semiclassical non-self-adjoint pseudodifferential operator

324 NILS DENCKER

pseudospectrum in [Pravda-Starov 2006a] we obtain a more refined view of the spectral instability seeDefinition 227 For example z is in the pseudospectrum of infinite index for P(h) if for any N theresolvent norm blows up faster than any power of the semiclassical parameter

(P(h)minus z Id)minus1 ge CN hminusN 0lt h 1 (11)

In [Dencker et al 2004] it was proved that (11) holds almost everywhere in the semiclassical pseu-dospectrum We shall generalize this to systems and prove that for systems of principal type exceptfor a nowhere dense set of degenerate values the resolvent blows up as in the scalar case see Theorem310 The complication is that the eigenvalues donrsquot have constant multiplicity in general only almosteverywhere

At the boundary of the semiclassical pseudospectrum we obtained in [Dencker et al 2004] a boundon the norm of the semiclassical resolvent under the additional condition of having no unbounded (orclosed) bicharacteristics In the systems case the picture is more complicated and it seems to be difficultto get an estimate on the norm of the resolvent using only information about the eigenvalues even in theprincipal type case see Example 41 In fact the norm is essentially preserved under multiplication withelliptic systems but the eigenvalues are changed Also the multiplicities of the eigenvalues could bechanging at all points on the boundary of the eigenvalues see Example 39 We shall instead introducequasisymmetrizable systems which generalize the normal forms of the scalar symbols at the boundaryof the eigenvalues see Definition 45 Quasisymmetrizable systems are of principal type and we obtainestimates on the resolvent as in the scalar case see Theorem 415

For boundary points of finite type we obtained in [Dencker et al 2004] subelliptic types of estimateson the semiclassical resolvent This is the case when one has nonvanishing higher order brackets Forsystems the situation is less clear there seems to be no general results on the subellipticity for systems Infact the real and imaginary parts do not commute in general making the bracket condition meaninglessEven when they do Example 52 shows that the bracket condition is not sufficient for subelliptic types ofestimates Instead we shall introduce invariant conditions on the order of vanishing of the symbol alongthe bicharacteristics of the eigenvalues For systems there could be several (limit) bicharacteristics of theeigenvalues going through a characteristic point see Example 59 Therefore we introduce the approx-imation property in Definition 510 which gives that the all (limit) bicharacteristics of the eigenvaluesare parallel at the characteristics see Remark 511 The general case presently looks too complicatedto handle We shall generalize the property of being of finite type to systems introducing systems ofsubelliptic type These are quasisymmetrizable systems satisfying the approximation property such thatthe imaginary part on the kernel vanishes of finite order along the bicharacteristics of the real part of theeigenvalues This definition is invariant under multiplication with invertible systems and taking adjointsand for these systems we obtain subelliptic types of estimates on the resolvent see Theorem 520

As an example we may look at

P(h)= h21 IdN +i K (x)

where 1 = minussumn

j=1 part2x j

is the positive Laplacian and K (x) isin Cinfin(Rn) is a symmetric N times N systemIf we assume some conditions of ellipticity at infinity for K (x) we may reduce to the case of boundedsymbols by Proposition 220 and Remark 221 see Example 222 Then we obtain that P(h) has discretespectrum in the right half plane z Re z ge 0 and in the first quadrant if K (x)ge 0 by Proposition 219

THE PSEUDOSPECTRUM OF SYSTEMS OF SEMICLASSICAL OPERATORS 325

We obtain from Theorem 310 that the L2 operator norm of the resolvent grows faster than any powerof h as h rarr 0 thus (11) holds for almost all values z such that Re z gt 0 and Im z is an eigenvalue of K see Example 312

For Re z = 0 and almost all eigenvalues Im z of K we find from Theorem 520 that the norm of theresolvent is bounded by Chminus23 see Example 522 In the case K (x) ge 0 and K (x) is invertible atinfinity we find from Theorem 415 that the norm of the resolvent is bounded by Chminus1 for Re z gt 0 andIm z = 0 by Example 417 The results in this paper are formulated for operators acting on the trivialbundle over Rn But since our results are mainly local they can be applied to operators on sections offiber bundles

2 The definitions

We shall consider N times N systems of semiclassical pseudo-differential operators and use the Weyl quan-tization

Pw(x h Dx)u =1

(2π)n

intintT lowastRn

P( x + y

)ei〈xminusyξ〉u(y) dy dξ (21)

for matrix valued P isin Cinfin(T lowastRnL(CN CN )) We shall also consider the semiclassical operators

P(h)sim

infinsumj=0

h j Pwj (x h D) (22)

with Pj isin Cinfin

b (TlowastRnL(CN CN )) Here Cinfin

b is the set of Cinfin functions having all derivatives in Linfin

and P0 = σ(P(h)) is the principal symbol of P(h) The operator is said to be elliptic if the principalsymbol P0 is invertible and of principal type if P0 vanishes of first order on the kernel see Definition31 Since the results in the paper only depend on the principal symbol one could also have used theKohnndashNirenberg quantization because the different quantizations only differ in the lower order termsWe shall also consider operators with analytic symbols then we shall assume that Pj (w) are boundedand holomorphic in a tubular neighborhood of T lowastRn satisfying

Pj (z ζ ) le C0C j j j|Im(z ζ )| le 1C forall j ge 0 (23)

which will give exponentially small errors in the calculus here A is the norm of the matrix A But theresults hold for more general analytic symbols see Remarks 311 and 419 In the following we shalluse the notation w = (x ξ) isin T lowastRn

We shall consider the spectrum Spec P(h) which is the set of values λ such that the resolvent (P(h)minusλ IdN )

minus1 is a bounded operator here IdN is the identity in CN The spectrum of P(h) is essentiallycontained in the spectrum of the principal symbol P(w) which is given by

|P(w)minus λ IdN | = 0

where |A| is the determinant of the matrix A For example if P(w) = σ(P(h)) is bounded and z1 isnot an eigenvalue of P(w) for any w = (x ξ) (or a limit eigenvalue at infinity) then P(h)minus z1 IdN isinvertible by Proposition 219 When P(w) is an unbounded symbol one needs additional conditionssee for example Proposition 220 We shall mostly restrict our study to bounded symbols but we can

326 NILS DENCKER

reduce to this case if P(h)minus z1 IdN is invertible by considering

(P(h)minus z1 IdN )minus1(P(h)minus z2 IdN ) z2 6= z1

see Remark 221 But unless we have conditions on the eigenvalues at infinity this does not always givea bounded operator

Example 21 Let

P(ξ)=

) ξ isin R

Then 0 is the only eigenvalue of P(ξ) but

(P(ξ)minus z IdN )minus1

= minus1z(

1 ξz0 1

)and (Pw minus z IdN )

minus1 Pw = minuszminus1 Pw is unbounded for any z 6= 0

Definition 22 Let P isin Cinfin(T lowastRn) be an N times N system We denote the closure of the set of eigenvaluesof P by

6(P)= λ isin C existw isin T lowastRn |P(w)minus λ IdN | = 0

and the eigenvalues at infinity

6infin(P)=λ isin C existw j rarr infin exist u j isin CN

0 |P(w j )u j minus λu j ||u j | rarr 0 j rarr infin

which is closed in C

In fact that 6infin(P) is closed follows by taking a suitable diagonal sequence Observe that as in thescalar case we could have 6infin(P)=6(P) for example if P(w) is constant in one direction It followsfrom the definition that λ isin6infin(P) if and only if the resolvent is defined and bounded when |w| is largeenough

(P(w)minus λ IdN )minus1

le C |w| 1 (24)

In fact if (24) does not hold there would exist w j rarr infin such that (P(w j )minusλ IdN )minus1

rarr infin j rarr infinThus there would exist u j isin CN such that |u j | = 1 and P(w j )u j minus λu j rarr 0 On the contrary if (24)holds then |P(w)u minus λu| ge |u|C for any u isin CN and |w| 1

It is clear from the definition that 6infin(P) contains all finite limits of eigenvalues of P at infinity Infact if P(w j )u j = λ j u j |u j | = 1 w j rarr infin and λ j rarr λ then

P(w j )u j minus λu j = (λ j minus λ)u j rarr 0

Example 21 shows that in general 6infin(P) could be a larger set

Example 23 Let P(ξ) be given by Example 21 then 6(P) = 0 but 6infin(P) = C In fact for anyλ isin C we find

|P(ξ)uξ minus λuξ | = λ2 when uξ =t(ξ λ)

We have that |uξ | =radic

|λ|2 + ξ 2 rarr infin so |P(ξ)uξ minus λuξ ||uξ | rarr 0 when |ξ | rarr infin

For bounded symbols we get equality according to the following proposition

Proposition 24 If P isin Cinfin

b (TlowastRn) is an N times N system then 6infin(P) is the set of all limits of the

eigenvalues of P at infinity

Proof Since 6infin(P) contains all limits of eigenvalues of P at infinity we only have to prove theopposite inclusion Let λ isin 6infin(P) then by the definition there exist w j rarr infin and u j isin CN such that|u j | = 1 and |P(w j )u j minus λu j | = ε j rarr 0 Then we may choose N times N matrix A j such that A j = ε j

and P(w j )u j = λu j + A j u j thus λ is an eigenvalue of P(w j ) minus A j Now if A and B are N times Nmatrices and d(Eig(A)Eig(B)) is the minimal distance between the sets of eigenvalues of A and Bunder permutations then we have that d(Eig(A)Eig(B))rarr 0 when A minus B rarr 0 In fact a theoremof Elsner [1985] gives

d(Eig(A)Eig(B))le N (2 max(A B))1minus1NA minus B

Since the matrices P(w j ) are uniformly bounded we find that they have an eigenvalue micro j such that|micro j minusλ| le CNε

1Nj rarr 0 as j rarr infin thus λ= lim jrarrinfin micro j is a limit of eigenvalues of P(w) at infinity

One problem with studying systems P(w) is that the eigenvalues are not very regular in the parame-ter w generally they depend only continuously (and eigenvectors measurably) on w

Definition 25 For an N times N system P isin Cinfin(T lowastRn) we define

κP(w λ)= Dim Ker(P(w)minus λ IdN )

K P(w λ)= maxk part

jλ p(w λ)= 0 for j lt k

where p(w λ) = |P(w)minus λ IdN | is the characteristic polynomial We have κP le K P with equality forsymmetric systems but in general we need not have equality see Example 27 If

k(P)=(w λ) isin T lowastRn

times C K P(w λ)ge k

k ge 1

then empty =N+1(P)subeN (P)sube middot middot middot sube1(P) and we may define

⋃jgt1

part j (P)

where part j (P) is the boundary of j (P) in the relative topology of 1(P)

Clearly j (P) is a closed set for any j ge 1 By definition we find that the multiplicity K P ofthe zeros of |P(w) minus λ IdN | is locally constant on 1(P) 4(P) If P(w) is symmetric then κP =

Dim Ker(P(w)minusλ IdN ) also is constant on 1(P)4(P) This is not true in general see Example 39

Remark 26 We find that4(P) is closed and nowhere dense in1(P) since it is the union of boundariesof closed sets We also find that

(w λ) isin4(P)hArr (w λ) isin4(Plowast)

since |Plowastminus λ IdN | = |P minus λ IdN |

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

)where λ j (w) isin Cinfin j = 1 2 then

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

but κP equiv 1 on 1(P)

Example 28 Let

(0 1t 0

)t isin R

Then P(t) has the eigenvalues plusmnradic

t and κP equiv 1 on 1(P)

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

We have that 2(P)= (w1 0 0w1) w1 isin R and κP = 2 on 2(P)

Definition 210 Let π j be the projections

π1(w λ)= w and π2(w λ)= λ

Then we define for λ isin C the closed sets

6λ(P)= π1(1(P)capπminus1

2 (λ))= w |P(w)minus λ IdN | = 0

X (P)= π1 (4(P))sube T lowastRn

Remark 211 Observe that X (P) is nowhere dense in T lowastRn and P(w) has constant characteristicsnear w0 isin X (P) This means that |P(w)minus λ IdN | = 0 if and only if λ = λ j (w) for j = 1 k wherethe eigenvalues λ j (w) 6= λk(w) for j 6= k when |wminusw0| 1

In fact πminus11 (w) is a finite set for any w isin T lowastRn and since the eigenvalues are continuous functions of

the parameters the relative topology on1(P) is generated by πminus11 (ω)cap1(P) for open sets ωsub T lowastRn

Definition 212 For an N times N system P isin Cinfin(T lowastRn) we define the weakly singular eigenvalue set

6ws(P)= π2 (4(P))sube C

and the strongly singular eigenvalue set

6ss(P)=λ πminus1

2 (λ)cap1(P)sube4(P)

Remark 213 It is clear from the definition that 6ss(P)sube6ws(P) We have that 6ws(P)cup6infin(P) and6ss(P)cup6infin(P) are closed and 6ss(P) is nowhere dense

In fact if λ j rarr λ isin 6infin(P) then πminus12 (λ j )cap1(P) is contained in a compact set for j 1 which

then either intersects4(P) or is contained in4(P) Since4(P) is closed these properties are preservedin the limit

Also if λ isin 6ss(P) then there exists (w j λ j ) isin 4(P) such that λ j rarr λ as j rarr infin Since 4(P)is nowhere dense in 1(P) there exists (w jk λ jk) isin1(P) 4(P) converging to (w j λ j ) as k rarr infinThen 6(P) 6ss(P) 3 λ j j rarr λ so 6ss(P) is nowhere dense On the other hand it is possible that6ws(P)=6(P) by the following example

Example 214 Let P(w) be the system in Example 29 then we have

6ws(P)=6(P)= R

and 6ss(P) = empty In fact the eigenvalues coincide only when w2 = w3 = 0 and the eigenvalue λ = w1

is also attained at some point where w2 6= 0 If we multiply P(w) with w4 + iw5 we obtain that6ws(P)=6(P)= C If we set P(w1 w2)= P(0 w1 w2) we find that 6ss(P)=6ws(P)= 0

Lemma 215 Let P isin Cinfin(T lowastRn) be an N times N system If (w0 λ0) isin1(P) 4(P) then there exists aunique Cinfin function λ(w) so that (w λ) isin1(P) if and only if λ= λ(w) in a neighborhood of (w0 λ0)If λ0 isin 6(P) (6ws(P) cup6infin(P)) then there is λ(w) isin Cinfin such that (w λ) isin 1(P) if and only ifλ= λ(w) in a neighborhood of 6λ0(P)

We find from Lemma 215 that 1(P)4(P) is locally given as a Cinfin manifold over T lowastRn and thatthe eigenvalues λ j (w) isin Cinfin outside X (P) This is not true if we instead assume that κP is constant on1(P) see Example 28

Proof If (w0 λ0) isin1(P) 4(P) then

λrarr |P(w)minus λ IdN |

vanishes of exactly order k gt 0 on 1(P) in a neighborhood of (w0 λ0) so

partkλ |P(w0)minus λ IdN | 6= 0 for λ= λ0

Thus λ = λ(w) is the unique solution to partkminus1λ |P(w)minus λ IdN | = 0 near w0 which is Cinfin by the Implicit

Function TheoremIf λ0 isin6(P)(6ws(P)cup6infin(P)) then we obtain this in a neighborhood of any w0 isin6λ0(P)b T lowastRn

Using a Cinfin partition of unity we find by uniqueness that λ(w) isin Cinfin in a neighborhood of 6λ0(P)

Remark 216 Observe that if λ0 isin 6(P) (6ws(P) cup 6infin(P)) and λ(w) isin Cinfin satisfies |P(w) minus

λ(w) IdN | equiv 0 near 6λ0(P) and λ|6λ0 (P) = λ0 then we find by Sardrsquos Theorem that d Re λ and d Im λ

are linearly independent on the codimension 2 manifold 6micro(P) for almost all values micro close to λ0 Thusfor n = 1 we find that 6micro(P) is a discrete set for almost all values micro close to λ0

In fact since λ0 isin6infin(P) we find that 6micro(P)rarr6λ0(P) when microrarr λ0 so 6micro(P)= w λ(w)= micro

for |microminus λ0| 1

Definition 217 A Cinfin function λ(w) is called a germ of eigenvalues at w0 for the N times N system P isin

Cinfin(T lowastRn) if|P(w)minus λ(w) IdN | equiv 0 in a neighborhood of w0

330 NILS DENCKER

If this holds in a neighborhood of every point in ωb T lowastRn then we say that λ(w) is a germ of eigenvaluesfor P on ω

Remark 218 If λ0 isin 6(P) (6ss(P) cup 6infin(P)) then there exists w0 isin 6λ0(P) so that (w0 λ0) isin

1(P) 4(P) By Lemma 215 there exists a Cinfin germ λ(w) of eigenvalues at w0 for P such thatλ(w0) = λ0 If λ0 isin 6(P) (6ws(P) cup 6infin(P)) then there exists a Cinfin germ λ(w) of eigenvalueson 6λ0(P)

As in the scalar case we obtain that the spectrum is essentially discrete outside 6infin(P)

Proposition 219 Assume that the N times N system P(h) is given by (22) with principal symbol P isin

Cinfin

b (TlowastRn) Let be an open connected set satisfying

cap6infin(P)= empty and cap 6(P) 6= empty

Then (P(h)minus z IdN )minus1 0 lt h 1 z isin is a meromorphic family of operators with poles of finite

rank In particular for h sufficiently small the spectrum of P(h) is discrete in any such set Whencap6(P)= empty we find that contains no spectrum of Pw(x h D)

Proof We shall follow the proof of Proposition 33 in [Dencker et al 2004] If satisfies the assumptionsof the proposition then there exists C gt 0 such that

|(P(w)minus z IdN )minus1

| le C if z isin and |w|gt C (25)

In fact otherwise there would exist w j rarr infin and z j isin such that |(P(w j )minus z j IdN )minus1

| rarr infin j rarr infinThus there exists u j isin CN such that |u j | = 1 and P(w j )u j minus z j u j rarr 0 Since 6(P)b C we obtain afterpicking a subsequence that z j rarr z isincap6infin(P)= empty The assumption that cap6(p) 6= empty implies thatfor some z0 isin we have (P(w)minus z0 IdN )

minus1isin Cinfin

b Let χ isin Cinfin

0 (TlowastRn) 0 le χ(w) le 1 and χ(w) = 1

when |w| le C where C is given by (25) Let

R(w z)= χ(w)(P(w)minus z0 IdN )minus1

+ (1 minusχ(w))(P(w)minus z IdN )minus1

isin Cinfin

for z isin The symbolic calculus then gives

Rw(x h D z)(P(h)minus z IdN )= I + h B1(h z)+ K1(h z)

(P(h)minus z IdN )Rw(x h D z)= I + h B2(h z)+ K2(h z)

where K j (h z) are compact operators on L2(Rn) depending holomorphically on z vanishing for z = z0and B j (h z) are bounded on L2(Rn) j = 1 2 By the analytic Fredholm theory we conclude that(P(h)minus z IdN )

minus1 is meromorphic in z isin for h sufficiently small When cap6(P)= empty we can chooseR(w z)= (P(w)minusz IdN )

minus1 then K j equiv 0 for j = 1 2 and P(h)minusz IdN is invertible for small enough h

We shall show how the reduction to the case of bounded operator can be done in the systems casefollowing [Dencker et al 2004] Let m(w) be a positive function on T lowastRn satisfying

1 le m(w)le C〈wminusw0〉N m(w0) forall w w0 isin T lowastRn

for some fixed C and N where 〈w〉 = 1 + |w| Then m is an admissible weight function and we candefine the symbol classes P isin S(m) by

partαwP(w) le Cαm(w) forallα

Following [Dimassi and Sjostrand 1999] we then define the semiclassical operator P(h)= Pw(x h D)In the analytic case we require that the symbol estimates hold in a tubular neighborhood of T lowastRn

partαwP(w) le Cαm(Rew) for |Imw| le 1C forallα (26)

One typical example of an admissible weight function is m(x ξ)= (〈ξ〉2+ 〈x〉

p)Now we make the ellipticity assumption

Pminus1(w) le C0mminus1(w) |w| 1 (27)

and in the analytic case we assume this in a tubular neighborhood of T lowastRn as in (26) By Leibnizrsquo rulewe obtain that Pminus1

isin S(mminus1) at infinity that is

partαwPminus1(w) le C prime

αmminus1(w) |w| 1

When z 6isin6(P)cup6infin(P) we find as before that

(P(w)minus z IdN )minus1

le C forallw

since the resolvent is uniformly bounded at infinity This implies that P(w)(P(w) minus z IdN )minus1 and

(P(w)minus z IdN )minus1 P(w) are bounded Again by Leibnizrsquo rule (27) holds with P replaced by P minus z IdN

thus (P(w)minusz IdN )minus1

isin S(mminus1) and we may define the semiclassical operator ((Pminusz IdN )minus1)w(x h D)

Since m ge 1 we find that P(w)minus z IdN isin S(m) so by using the calculus we obtain that

(Pw minus z IdN )((P minus z IdN )minus1)w = 1 + h Rw1

((P minus z IdN )minus1)w(Pw minus z IdN )= 1 + h Rw2

where R j isin S(1) j = 1 2 For small enough h we get invertibility and the following result

Proposition 220 Assume that P isin S(m) is an NtimesN system satisfying (27) and that z 6isin6(P)cup6infin(P)Then we find that Pw minus z IdN is invertible for small enough h

This makes it possible to reduce to the case of operators with bounded symbols

Remark 221 If z1 isin Spec(P) we may define the operator

Q = (P minus z1 IdN )minus1(P minus z2 IdN ) z2 6= z1

The resolvents of Q and P are related by

(Q minus ζ IdN )minus1

= (1 minus ζ )minus1(P minus z1 IdN )(

P minusζ z1 minus z2

ζ minus 1IdN

)minus1ζ 6= 1

when (ζ z1 minus z2)(ζ minus 1) isin Spec(P)

332 NILS DENCKER

Example 222 LetP(x ξ)= |ξ |2 IdN +i K (x)

where 0 le K (x) isin Cinfin

b then we find that P isin S(m) with m(x ξ)= |ξ |2 +1 If 0 isin6infin(K ) then K (x) isinvertible for |x | 1 so Pminus1

isin S(mminus1) at infinity Since Re z ge 0 in6(P) we find from Proposition 220that Pw(x h D)+ IdN is invertible for small enough h and Pw(x h D)(Pw(x h D)+ IdN )

minus1 is boundedin L2 with principal symbol P(w)(P(w)+ IdN )

minus1isin Cinfin

In order to measure the singularities of the solutions we shall introduce the semiclassical wave frontsets

Definition 223 For u isin L2(Rn) we say that w0 isin WFh(u) if there exists a isin Cinfin

0 (TlowastRn) such that

a(w0) 6= 0 and the L2 normaw(x h D)u le Ckhk

forall k (28)

We call WFh(u) the semiclassical wave front set of u

Observe that this definition is equivalent to Definition (25) in [Dencker et al 2004] which use the FBItransform T given by (426) w0 isin WFh(u) if T u(w)= O(hinfin)when |wminusw0| 1 We may also definethe analytic semiclassical wave front set by the condition that T u(w) = O(eminusch) in a neighborhoodof w0 for some c gt 0 see (26) in [Dencker et al 2004]

Observe that if u = (u1 uN )isin L2(RnCN )we may define WFh(u)=⋂

j WFh(u j ) but this gives noinformation about which components of u that are singular Therefore we shall define the correspondingvector valued polarization sets

Definition 224 For u isin L2(RnCN ) we say that (w0 z0) isin WFpolh (u)sube T lowastRn

timesCN if there exists A(h)given by (22) with principal symbol A(w) such that A(w0)z0 6= 0 and A(h)u satisfies (28) We callWFpol

h (u) the semiclassical polarization set of u

We could similarly define the analytic semiclassical polarization set by using the FBI transform andanalytic pseudodifferential operators

Remark 225 The semiclassical polarization sets are closed linear in the fiber and has the functorialproperties of the Cinfin polarization sets in [Dencker 1982] In particular we find that

π(WFpolh (u) 0)= WFh(u)=

WFh(u j )

if π is the projection along the fiber variables π T lowastRntimes CN

7rarr T lowastRn We also find that

A(WFpolh (u))=

(w A(w)z) (w z) isin WFpol

sube WFpolh (A(h)u)

if A(w) is the principal symbol of A(h) which implies that WFpolh (Au) = A(WFpol

h (u)) when A(h) iselliptic

This follows from the proofs of Propositions 25 and 27 in [Dencker 1982]

Example 226 Let u = (u1 uN ) isin L2(T lowastRnCN ) where WFh(u1) = w0 and WFh(u j ) = empty forj gt 1 Then

WFpolh (u)= (w0 (z 0 )) z isin C

since Aw(x h D)u = O(hinfin) if Awu =sum

jgt1 Awj u j and w0 isin WFh(u) By taking a suitable invertibleE we obtain

WFpolh (Eu)= (w0 zv) z isin C

for any v isin CN

We shall use the following definitions from [Pravda-Starov 2006a] here and in the following P(h)will denote the L2 operator norm of P(h)

Definition 227 Let P(h) 0lt h le 1 be a semiclassical family of operators on L2(Rn) with domain DFor micro gt 0 we define the pseudospectrum of index micro as the set

3scmicro (P(h))=

z isin C forall C gt 0 forall h0 gt 0 exist 0lt h lt h0 (P(h)minus z IdN )

minus1 ge Chminusmicro

and the injectivity pseudospectrum of index micro as

λscmicro (P(h)) =

z isin C forall C gt 0 forall h0 gt 0 exist 0lt h lt h0 exist u isin D u = 1 (P(h)minus z IdN )u le Chmicro

We define the pseudospectrum of infinite index as 3scinfin(P(h))=

⋂micro3

scmicro (P(h)) and correspondingly the

injectivity pseudospectrum of infinite index

Here we use the convention that (P(h)minus λ IdN )minus1

= infin when λ is in the spectrum Spec(P(h))Observe that we have the obvious inclusion λsc

micro (P(h)) sube 3scmicro (P(h)) for all micro We get equality if for

example P(h) is Fredholm of index ge 0

3 The interior case

Recall that the scalar symbol p(x ξ) isin Cinfin(T lowastRn) is of principal type if dp 6= 0 when p = 0 In thefollowing we let partνP(w) = 〈ν d P(w)〉 for P isin C1(T lowastRn) and ν isin T lowastRn We shall use the followingdefinition of systems of principal type in fact most of the systems we consider will be of this type Weshall denote Ker P and Ran P the kernel and range of P

Definition 31 The N times N system P(w) isin Cinfin(T lowastRn) is of principal type at w0 if

Ker P(w0) 3 u 7rarr partνP(w0)u isin Coker P(w0)= CNRan P(w0) (31)

is bijective for some ν isin Tw0(TlowastRn) The operator P(h) given by (22) is of principal type if the principal

symbol P = σ(P(h)) is of principal type

Remark 32 If P(w) isin Cinfin is of principal type and A(w) B(w) isin Cinfin are invertible then AP B is ofprincipal type We have that P(w) is of principal type if and only if the adjoint Plowast is of principal type

In fact by Leibnizrsquo rule we have

part(AP B)= (partA)P B + A(partP)B + APpartB (32)

and Ran(AP B) = A(Ran P) and Ker(AP B) = Bminus1(Ker P) when A and B are invertible which givesthe invariance under left and right multiplication Since Ker Plowast(w0) = Ran P(w0)

perp we find that Psatisfies (31) if and only if

Ker P(w0)times Ker Plowast(w0) 3 (u v) 7rarr 〈partνP(w0)u v〉

334 NILS DENCKER

is a nondegenerate bilinear form Since 〈partνPlowastv u〉 = 〈partνPu v〉 we find that Plowast is of principal type ifand only if P is

Observe that if P only has one vanishing eigenvalue λ (with multiplicity one) then the condition thatP is of principal type reduces to the condition in the scalar case dλ 6= 0 In fact by using the spectralprojection one can find invertible systems A and B so that

AP B =

(λ 00 E

)with E invertible (N minus 1)times (N minus 1) system and this system is obviously of principal type

Example 33 Consider the system in Example 27

(λ1(w) 1

0 λ2(w)

)where λ j (w)isin Cinfin j = 1 2 We find that P(w)minusλ Id2 is not of principal type when λ= λ1(w)= λ2(w)

since Ker(P(w)minus λ Id2)= Ran(P(w)minus λ Id2)= C times 0 is preserved by partP

Observe that the property of being of principal type is not stable under C1 perturbation not even whenP = Plowast is symmetric by the following example

Example 34 The system

(w1 minusw2 w2

w2 minusw1 minusw2

)= Plowast(w) w = (w1 w2)

is of principal type when w1 = w2 = 0 but not of principal type when w2 6= 0 and w1 = 0 In fact

partw1 P =

(1 00 minus1

)is invertible and when w2 6= 0 we have that

Ker P(0 w2)= Ker partw2 P(0 w2)= z(1 1) z isin C

which is mapped to Ran P(0 w2)= z(1minus1) z isin C by partw1 P

We shall obtain a simple characterization of systems of principal type Recall κP K P and4(P) givenby Definition 25

Proposition 35 Assume P(w) isin Cinfin is an N times N system and that (w0 λ0) isin 1(P) 4(P) thenP(w)minus λ0 IdN is of principal type at w0 if and only if κP equiv K P at (w0 λ0) and dλ(w0) 6= 0 for the Cinfin

germ of eigenvalues λ(w) for P at w0 satisfying λ(w0)= λ0

Thus in the case λ0 = 0 isin 6ws(P) we find that P(w) is of principal type if and only if λ is ofprincipal type and we have no nontrivial Jordan boxes in the normal form Observe that by the proof ofLemma 215 the Cinfin germ λ(w) is the unique solution to partk

λ p(w λ) = 0 for k = K P(w λ)minus 1 wherep(w λ) = |P(w)minus λ IdN | is the characteristic equation Thus we find that dλ(w) 6= 0 if and only ifpartwpart

kλ p(w λ) 6= 0 For symmetric operators we have κP equiv K P and we only need this condition when

(w0 λ0) isin4(P)

Example 36 The system P(w) in Example 34 has eigenvalues minusw2 plusmn

w21 +w2

2 which are equalif and only if w1 = w2 = 0 so 0 = 6ws(P) When w2 6= 0 and w1 asymp 0 the eigenvalue close tozero is w2

12w2 + O(w41) which has vanishing differential at w1 = 0 The characteristic equation is

p(w λ)= λ2+ 2λw2 minusw2

1 so dw p = 0 when w1 = λ= 0

Proof of Proposition 35 Of course it is no restriction to assume λ0 = 0 First we note that P(w) is ofprincipal type at w0 if and only if

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

for some ν isin T (T lowastRn) Observe that part j|P(w0)| = 0 for j lt k In fact by choosing bases for Ker P(w0)

and Im P(w0) respectively and extending to bases of CN we obtain matrices A and B so that

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

)where |P22(w0)| 6= 0 and P11 P12 and P21 all vanish at w0 By the invariance P is of principal type ifand only if partνP11 is invertible for some ν so by expanding the determinant we obtain (33)

Since (w0 0) isin 1(P) 4(P) we find from Lemma 215 that we may choose a neighborhood ω of(w0 0) such that (w λ) isin1(P)capω if and only if λ= λ(w) isin Cinfin Then

|P(w)minus λ IdN | = (λ(w)minus λ)me(w λ)

nearw0 where e(w λ) 6= 0 and m = K P(w0 0)ge κP(w0 0) Letting λ= 0 we obtain that part jν |P(w0)|= 0

if j lt m and partmν |P(w0)| = (partνλ(w0))

me(w0 0)

Remark 37 Proposition 35 shows that for a symmetric system the property to be of principal type isstable outside4(P) if the symmetric system P(w)minusλ IdN is of principal type at a point (w0 λ0) isin4(P)then it is in a neighborhood It follows from Sardrsquos Theorem that symmetric systems P(w)minus λ IdN areof principal type almost everywhere on 1(P)

In fact for symmetric systems we have κP equiv K P and the differential dλ 6= 0 almost everywhere on1(P) 4(P) For Cinfin germs of eigenvalues we can define the following bracket condition

Definition 38 Let P isin Cinfin(T lowastRn) be an N times N system then we define

3(P)=3minus(P)cup3+(P)

where 3plusmn(P) is the set of λ0 isin6(P) such that there exists w0 isin6λ0(P) so that (w0 λ0) isin4(P) and

plusmn Re λ Im λ (w0) gt 0 (34)

for the unique Cinfin germ λ(w) of eigenvalues at w0 for P such that λ(w0)= λ0

Observe that 3plusmn(P) cap 6ss(P) = empty and it follows from Proposition 35 that P(w) minus λ0 IdN is ofprincipal type at w0 isin 3plusmn(P) if and only if κP = K P at (w0 λ0) since dλ(w0) 6= 0 when (34) holdsBecause of the bracket condition (34) we find that3plusmn(P) is contained in the interior of the values6(P)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

)(x ξ) isin T lowastR

336 NILS DENCKER

where q(x ξ) = ξ + i x2 and 0 le χ(x) isin Cinfin(R) such that χ(x) = 0 when x le 0 and χ(x) gt 0when x gt 0 Then 6(P) = Im z ge 0 3plusmn(P) = Im z gt 0 and 4(P) = empty For Im λ gt 0 we find6λ(P)= (plusmn

radicIm λRe λ) and P minus λ Id2 is of principal type at 6λ(P) only when x lt 0

Theorem 310 Let P isin Cinfin(T lowastRn) be an N times N system then we have that

3(P) (6ws(P)cup6infin(P)

)sube3minus(P) (35)

when n ge 2 Assume that P(h) is given by (22) with principal symbol P isin Cinfin

b (TlowastRn) and that λ0 isin

3minus(P) 0 6= u0 isin Ker(P(w0)minus λ0 IdN ) and P(w)minus λ IdN is of principal type on 6λ(P) near w0 for|λ minus λ0| 1 for the w0 isin 6λ0(P) in Definition 38 Then there exists h0 gt 0 and u(h) isin L2(Rn)0lt h le h0 so that u(h) le 1

(P(h)minus λ0 IdN )u(h) le CN hNforall N 0lt h le h0 (36)

and WFpolh (u(h))= (w0 u0) There also exists a dense subset of values λ0 isin3(P) so that

(P(h)minus λ0 IdN )minus1

ge C prime

N hminusNforall N (37)

If all the terms Pj in the expansion (22) are analytic satisfying (23) then hplusmnN may be replaced byexp(∓ch) in (36)ndash(37)

= infin when λ is in the spectrum Spec(P(h))Condition (36) means that λ0 is in the injectivity pseudospectrum λsc

infin(P) and (37) means that λ0 is in

the pseudospectrum 3scinfin(P)

Remark 311 If P(h) is Fredholm of nonnegative index then (36) holds for λ0 in a dense subset of3(P) In the analytic case it follows from the proof that it suffices that Pj (w) is analytic satisfying (23)in a fixed complex neighborhood of w0 isin6λ(P) for all j

In fact if P(h) is Fredholm of nonnegative index and λ0 isin Spec(P(h)) then the dimension ofKer(P(h)minus λ0 IdN ) is positive and (36) holds

Example 312 LetP(x ξ)= |ξ |2 Id +i K (x) (x ξ) isin T lowastRn

where K (x) isin Cinfin(Rn) is symmetric for all x Then we find that

3minus(P)=3(P)=

Re z ge 0 and Im z isin6(K ) (6ss(K )cup6infin(K )

In fact for any Im z isin6(K ) (6ss(K )⋃6infin(K )) there exists a germ of eigenvalues λ(x) isin Cinfin(ω) for

K (x) in an open set ω sub Rn so that λ(x0) = Im z for some x0 isin ω By Sardrsquos Theorem we find thatalmost all values of λ(x) in ω are nonsingular and if dλ 6= 0 and Re z gt 0 we may choose ξ0 isin Rn sothat |ξ0|

2= Re z and 〈ξ0 partxλ〉 lt 0 Then the Cinfin germ of eigenvalues |ξ |2 + iλ(x) for P satisfies (34)

at (x0 ξ0) with the minus sign Since K (x) is symmetric we find that P(w)minus z IdN is of principal type

Proof of Theorem 310 First we are going to prove (35) in the case n ge 2 Let

W =6ws(P)cup6infin(P)

which is a closed set by Remark 213 then we find that every point in 3(P) W is a limit point of(3minus(P)cup3+(P)

) W = (3minus(P) W )cup (3+(P) W )

Thus we only have to show that λ0 isin3minus(P) if

λ0 isin3+(P) W =3+(P) (6ws(P)cup6infin(P)

) (38)

By Lemma 215 and Remark 216 we find from (38) that there exists a Cinfin germ of eigenvalues λ(w) isinCinfin so that 6micro(P) is equal to the level sets w λ(w)= micro for |microminusλ0| 1 By definition we find thatthe Poisson bracket Re λ Im λ does not vanish identically on 6λ0(P) Now by Remark 216 d Re λand d Im λ are linearly independent on 6micro(P) for almost all micro close to λ0 and then 6micro(P) is a Cinfin

manifold of codimension 2 By using Lemma 31 of [Dencker et al 2004] we obtain that Re λ Im λ

changes sign on 6micro(P) for almost all values micro near λ0 so we find that those values also are in 3minus(P)By taking the closure we obtain (35)

Next assume that λisin3minus(P) it is no restriction to assume λ=0 By the assumptions there existsw0 isin

60(P) and λ(w)isin Cinfin such that λ(w0)= 0 Re λ Im λlt 0 at w0 (w0 0) isin4(P) and P(w)minusλ IdN isof principal type on 6λ(P) near w0 when |λ| 1 Then Proposition 35 gives that κP equiv K P is constanton 1(P) near (w0 λ0) so

Dim Ker(P(w)minus λ(w) IdN )equiv K gt 0

in a neighborhood of w0 Since the dimension is constant we can construct a base u1(w) uK (w) isin

Cinfin for Ker(P(w)minus λ(w) IdN ) in a neighborhood of w0 By orthonormalizing it and extending to CN

we obtain orthogonal E(w) isin Cinfin so that

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

If P(w) is analytic in a tubular neighborhood of T lowastRn then E(w) can be chosen analytic in that neigh-borhood Since P0 is of principal type at w0 by Remark 32 and partP0(w0) maps Ker P0(w0) into itselfwe find that Ran P0(w0)cap Ker P0(w0)= 0 and thus |P22(w0)| 6= 0 In fact if there exists uprimeprime

6= 0 suchthat P22(w0)uprimeprime

= 0 then by applying P(w0) on u = (0 uprimeprime) isin Ker P0(w0) we obtain

0 6= P0(w0)u = (P12(w0)uprimeprime 0) isin Ker P0(w0)cap Ran P0(w0)

which gives a contradiction Clearly the norm of the resolvent P(h)minus1 only changes with a multiplica-tive constant under left and right multiplication of P(h) by invertible systems Now Ew(x h D) and(Elowast)w(x h D) are invertible in L2 for small enough h and

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

)where σ(P11)= λ IdN P21 = O(h) and P22(h) is invertible for small h By multiplying from the right by(

IdK 0minusP22(h)minus1 P21(h) IdNminusK

338 NILS DENCKER

for small h we obtain that P21(h) equiv 0 and this only changes lower order terms in P11(h) Then bymultiplying from the left by (

IdK minusP12(h)P22(h)minus1

0 IdNminusK

)we obtain that P12(h)equiv 0 without changing P11(h) or P22(h)

Thus in order to prove (36) we may assume N = K and P(w)= λ(w) IdK By conjugating similarlyas in the scalar case (see the proof of Proposition 2631 in Volume IV of [Hormander 1983ndash1985]) wecan reduce to the case when P(h)= λw(x h D) IdK In fact let

P(h)= λw(x h D) IdK +

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

jge0 h j Awj (x h D) and B(h) =sum

jge0 h j Bwj (x h D) with B0(w) equiv A0(w) Then the calculusgives

P(h)A(h)minus B(h)λw(x h D)=

sumjge1

h j Ewj (x h D)

Ek =12i

Hλ(Akminus1 + Bkminus1)+ P1 Akminus1 + λ(Ak minus Bk)+ Rk k ge 1

Here Hλ is the Hamilton vector field of λ Rk only depends on A j and B j for j lt k minus1 and R1 equiv 0 Nowwe can choose A0 so that A0 = IdK on V0 = w Im λ(w)= 0 and 1

i HλA0 + P1 A0 vanishes of infiniteorder on V0 near w0 In fact since Re λ Im λ 6= 0 we find d Im λ 6= 0 on V0 and V0 is noncharacteristicfor HRe λ Thus the equation determines all derivatives of A0 on V0 and we may use the Borel Theoremto obtain a solution Then by taking

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

we obtain E0 equiv 0 Lower order terms are eliminated similarly by making

Hλ(A jminus1 + B jminus1)+ P1 A jminus1 + R j

vanish of infinite order on V0 Observe that only the difference A jminus1minusB jminus1 is determined in the previousstep Thus we can reduce to the case P = λw(x h D) Id and then the Cinfin result follows from the scalarcase (see Theorem 12 in [Dencker et al 2004]) by using Remark 225 and Example 226

The analytic case follows as in the proof of Theorem 12prime in [Dencker et al 2004] by applying aholomorphic WKB construction to P = P11 on the form

u(z h)sim eiφ(z)hinfinsumj=0

A j (z)h j z = x + iy isin Cn

which will be an approximate solution to P(h)u(z h) = 0 Here P(h) is given by (22) with P0(w) =

λ(w) Id Pj satisfying (23) and Pwj (z h Dz) given by the formula (21) where the integration may bedeformed to a suitable chosen contour instead of T lowastRn (see [Sjostrand 1982 Section 4]) The holo-morphic phase function φ(z) satisfying λ(z dzφ) = 0 is constructed as in [Dencker et al 2004] so that

dzφ(x0) = ξ0 and Imφ(x) ge c|x minus x0|2 c gt 0 and w0 = (x0 ξ0) The holomorphic amplitude A0(z)

satisfies the transport equationsumj

partζ jλ(z dzφ(z))Dz j A0(z)+ P1(z dzφ(z))A0(z)= 0

with A0(x0) 6= 0 The lower order terms in the expansion solvesumj

partζ jλ(z dzφ(z))Dz j Ak(z)+ P1(z dzφ(z))Ak(z)= Sk(z)

where Sk(z) only depends on A j and Pj+1 for j lt k As in the scalar case we find from (23) that thesolutions satisfy Ak(z) le C0Ckkk see Theorem 93 in [Sjostrand 1982] By solving up to k lt chcutting of near x0 and restricting to Rn we obtain that P(h)u = O(eminusch) The details are left to thereader see the proof of Theorem 12prime in [Dencker et al 2004]

For the last result we observe thatRe λ Im λ

= minusRe λ Im λ λ isin 6(P) hArr λ isin 6(Plowast) Plowast is

of principal type if and only if P is and Remark 26 gives (w λ) isin 4(P) hArr (w λ) isin 4(Plowast) Thusλ isin3+(P) if and only if λ isin3minus(Plowast) and

(P(h)minus λ IdN )minus1

= (Plowast(h)minus λ IdN )minus1

From the definition we find that any λ0 isin 3(P) is an accumulation point of 3plusmn(P) so we obtain theresult from (36)

Remark 313 In order to get the estimate (36) it suffices that there exists a semibicharacteristic 0of λminus λ0 through w0 such that 0 times λ0 cap4(P) = empty P(w)minus λ IdN is of principal type near 0 forλ near λ0 and that condition (9) is not satisfied on 0 see [Hormander 1983ndash1985 Definition 2646Volume IV] This means that there exists 0 6= q isin Cinfin such that 0 is a bicharacteristic of Re q(λminus λ0)

through w0 and Im q(λminus λ0) changes sign from + to minus when going in the positive direction on 0

In fact once we have reduced to the normal form (310) the construction of approximate local solu-tions in the proof of [Hormander 1983ndash1985 Theorem 2647 Volume IV] can be adapted to this casesince the principal part is scalar See also Theorem 13 in [Pravda-Starov 2006b Section 32] for a similarscalar semiclassical estimate

When P(w) is not of principal type the reduction in the proof of Theorem 310 may not be possiblesince P22 in (39) needs not be invertible by the following example

Example 314 Let

(λw(x h D) 1

h λw(x h D)

)where λ isin Cinfin satisfies the bracket condition (34) The principal symbol is

(λ(w) 1

0 λ(w)

)with eigenvalue λ(w) and we have

Ker(P(w)minus λ(w) Id2)= Ran(P(w)minus λ(w) Id2)= (z 0) z isin C forallw

340 NILS DENCKER

We find that P is not of principal type since d P = dλ Id2 Observe that 4(P)= empty since K P is constanton 1(P)

When the dimension is equal to one we have to add some conditions in order to get the inclusion (35)

Lemma 315 Let P(w)isin Cinfin(T lowastR) be an N timesN system Then for every component of C(6ws(P)cup6infin(P)) which has nonempty intersection with 6(P) we find that

sube3minus(P) (311)

The condition of having nonempty intersection with the complement is necessary even in the scalarcase see the remark and Lemma 32rsquo on page 394 in [Dencker et al 2004]

Proof If micro isin6infin(P) we find that the index

i = var argγ |P(w)minusmicro IdN | (312)

is well-defined and continuous when γ is a positively oriented circle w |w| = R for R 1 If λ0 isin

6ws(P)cup6infin(P) then we find from Lemma 215 that the characteristic polynomial is equal to

|P(w)minusmicro IdN | = (λ(w)minusmicro)ke(wmicro)

near w0 isin 6λ0(P) here λ e isin Cinfin e 6= 0 and k = K P(w0) By Remark 216 we find for almost all microclose to λ0 that d Re λandd Im λ 6= 0 on λminus1(micro)=6micro(P) which is then a finite set of points on which thePoisson bracket is nonvanishing If micro isin6(P) we find that the index (312) vanishes since one can thenlet R rarr 0 Thus if a component of C (6ws(P)cup6infin(P)) has nonempty intersection with 6(P)we obtain that i = 0 in When micro0 isin cap3(P) we find from the definition that the Poisson bracketRe λ Im λ cannot vanish identically on 6micro(P) for all micro close to micro0 Since the index is equal to thesum of positive multiples of the values of the Poisson brackets at 6micro(P) we find that the bracket mustbe negative at some point w0 isin6micro(P) for almost all micro near λ0 which gives (311)

4 The quasisymmetrizable case

First we note that if the system P(w)minus z IdN is of principal type near 6z(P) for z close to λ isin part6(P)(6ws(P)cup6infin(P)) and6λ(P) has no closed semibicharacteristics then one can generalize Theorem 13in [Dencker et al 2004] to obtain

le Ch h rarr 0 (41)

In fact by using the reduction in the proof of Theorem 310 this follows from the scalar case see Example412 But then the eigenvalues close to λ have constant multiplicity

Generically we have that the eigenvalues of the principal symbol P have constant multiplicity almosteverywhere since 4(P) is nowhere dense But at the boundary part6(P) this needs not be the case Forexample if

P(t τ )= τ Id +i K (t)

where Cinfin3 K ge 0 is unbounded and 0 isin6ss(K ) then R = part6(P)sube6ss(P)

When the multiplicity of the eigenvalues of the principal symbol is not constant the situation is morecomplicated The following example shows that then it is not sufficient to have conditions only on theeigenvalues in order to obtain the estimate (41) not even in the principal type case

Example 41 Let a1(t) a2(t) isin Cinfin(R) be real valued a2(0)= 0 aprime

2(0) gt 0 and let

Pw(t h Dt)=

(h Dt + a1(t) a2(t)minus ia1(t)

a2(t)+ ia1(t) minush Dt + a1(t)

)= Pw(t h Dt)

lowast

Then the eigenvalues of P(t τ ) are

λ= a1(t)plusmnradicτ 2

+ a21(t)+ a2

2(t) isin R

which coincide if and only if τ = a1(t)= a2(t)= 0 We have that

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

2a1(t) h Dt minus ia2(t)

)= P(h)

Thus we can construct uh(t) =t(0 u2(t)) so that uh = 1 and P(h)uh = O(hN ) for h rarr 0 see

Theorem 12 in [Dencker et al 2004] When a2 is analytic we may obtain that P(h)uh = O(exp(minusch))by Theorem 12prime in [Dencker et al 2004] By the invariance we see that P is of principal type at t = τ =0if and only if a1(0)= 0 If a1(0)= 0 then6ws(P)=0 and when a1 6= 0 we have that Pw is a self-adjointdiagonalizable system In the case a1(t) equiv 0 and a2(t) equiv t the eigenvalues of P(t h Dt) are plusmn

radic2nh

n isin N see the proof of Proposition 361 in [Helffer and Sjostrand 1990]

Of course the problem is that the eigenvalues are not invariant under multiplication with ellipticsystems To obtain the estimate (41) for operators that are not of principal type it is not even sufficientthat the eigenvalues are real having constant multiplicity

Example 42 Let a(t) isin Cinfin(R) be real valued a(0)= 0 aprime(0) gt 0 and

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

Then the principal symbol is

P(t τ )=

(τ a(t)0 τ

)so the only eigenvalue is τ Thus 4(P) = empty but the principal symbol is not diagonalizable and whena(t) 6= 0 the system is not of principal type We have(

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

)thus we obtain that Pw(t h Dt)

minus1 ge CN hminusN for all N when h rarr 0 by using Example 41 with a1 equiv 0

and a2 equiv a When a is analytic we obtain P(t h Dt)minus1

ge exp(cradic

For nonprincipal type operators to obtain the estimate (41) it is not even sufficient that the principalsymbol has real eigenvalues of multiplicity one

342 NILS DENCKER

Example 43 Let a(t) isin Cinfin(R) a(0)= 0 aprime(0) gt 0 and

(1 h Dt

h iha(t)

)with principal symbol (

)thus the eigenvalues are 0 and 1 so 4(P)= empty Since(

1 0minush 1

(1 minush Dt

(1 00 minush

) (1 00 h Dt minus ia(t)

)we obtain as in Example 41 that P(h)minus1

ge CN hminusN when h rarr 0 for all N and for analytic a(t)we obtain P(h)minus1

ge Cech h rarr 0 Now partτ P maps Ker P(0) into Ran P(0) so the system is notof principal type Observe that this property is not preserved under the multiplications above since thesystems are not elliptic

Instead of using properties of the eigenvalues of the principal symbol we shall use properties thatare invariant under multiplication with invertible systems First we consider the scalar case recall that ascalar p isin Cinfin is of principal type if dp 6= 0 when p = 0 We have the following normal form for scalarprincipal type operators near the boundary part6(P) Recall that a semibicharacteristic of p is a nontrivialbicharacteristic of Re qp for q 6= 0

Example 44 Assume that p(x ξ) isin Cinfin(T lowastRn) is of principal type and 0 isin part6(p)6infin(p) Then wefind from the proof of Lemma 41 in [Dencker et al 2004] that there exists 0 6= q isin Cinfin so that

Im qp ge 0 and d Re qp 6= 0

in a neighborhood of w0 isin 60(p) In fact condition (17) in that lemma is not needed to obtain a localpreparation By making a symplectic change of variables and using the Malgrange preparation theoremwe then find that

p(x ξ)= e(x ξ)(ξ1 + i f (x ξ prime)) ξ = (ξ1 ξprime) (42)

in a neighborhood of w0 isin 60(p) where e 6= 0 and f ge 0 If there are no closed semibicharacteristicsof p then we obtain this in a neighborhood of 60(p) by a partition of unity

This normal form in the scalar case motivates the following definition

Definition 45 We say that the N times N system P(w) isin Cinfin(T lowastRn) is quasisymmetrizable with respectto the real Cinfin vector field V in sube T lowastRn if exist N times N system M(w) isin Cinfin(T lowastRn) so that in we have

Re〈M(V P)u u〉 ge cu2minus CPu

2 c gt 0 (43)

Im〈M Pu u〉 ge minusCPu2 (44)

for any u isin CN the system M is called a symmetrizer for P

The definition is clearly independent of the choice of coordinates in T lowastRn and choice of base in CN When P is elliptic we may take M = i Plowast as multiplier then P is quasisymmetrizable with respect toany vector field because Pu sim= u Observe that for a fixed vector field V the set of multipliers Msatisfying (43)ndash(44) is a convex cone a sum of two multipliers is also a multiplier Thus given a vectorfield V it suffices to make a local choice of multiplier M and then use a partition of unity to get a globalone

Taylor has studied symmetrizable systems of the type Dt Id +i K for which there exists Rgt 0 makingRK symmetric (see Definition 432 in [Taylor 1981]) These systems are quasisymmetrizable withrespect to partτ with symmetrizer R We see from Example 44 that the scalar symbol p of principal typeis quasisymmetrizable in neighborhood of any point at part6(p) 6infin(p)

The invariance properties of quasisymmetrizable systems are partly due to the following simple andprobably well-known result on semibounded matrices In the following we shall denote Re A =

12(A +

Alowast) and i Im A =12(A minus Alowast) the symmetric and antisymmetric parts of the matrix A Also if U and V

are linear subspaces of CN then we let U + V = u + v u isin U and v isin V

Lemma 46 Assume that Q is an N times N matrix such that Im zQ ge 0 for some 0 6= z isin C Then we find

Ker Q = Ker Qlowast= Ker(Re Q)cap Ker(Im Q) (45)

and Ran Q = Ran(Re Q)+ Ran(Im Q) is orthogonal to Ker Q

Proof By multiplying with z we may assume that Im Q ge 0 clearly the conclusions are invariant undermultiplication with complex numbers If u isinKer Q then we have 〈Im Qu u〉= Im〈Qu u〉=0 By usingthe CauchyndashSchwarz inequality on Im Q ge 0 we find that 〈Im Qu v〉 = 0 for any v Thus u isin Ker(Im Q)so Ker Q sube Ker Qlowast We get equality and (45) by the rank theorem since Ker Qlowast

= Ran QperpFor the last statement we observe that Ran Q sube Ran(Re Q)+ Ran(Im Q)= (Ker Q)perp by (45) where

we also get equality by the rank theorem

Proposition 47 Assume that P(w) isin Cinfin(T lowastRn) is a quasisymmetrizable system then we find that Pis of principal type Also the symmetrizer M is invertible if Im M P ge cPlowast P for some c gt 0

Observe that by adding iPlowast to M we may assume that Q = M P satisfies

Im Q ge (minus C)Plowast P ge Plowast P ge cQlowastQ c gt 0 (46)

for ge C + 1 and then the symmetrizer is invertible by Proposition 47

Proof Assume that (43)ndash(44) hold at w0 Ker P(w0) 6= 0 but (31) is not a bijection Then there exists0 6= u isin Ker P(w0) and v isin CN such that V P(w0)u = P(w0)v so (43) gives

Re〈M P(w0)v u〉 = Re〈MV P(w0)u u〉 ge cu2 gt 0

This means thatRan M P(w0) 6sube Ker P(w0)

perp (47)

Let M = M + iPlowast then we have that

Im〈MPu u〉 ge (minus C)Pu2

344 NILS DENCKER

so for large enough we have Im MP ge 0 By Lemma 46 we find

Ran MPperp Ker MP

Since Ker P sube Ker MP and Ran Plowast P sube Ran Plowastperp Ker P we find that Ran M Pperp Ker P for any This

gives a contradiction to (47) thus P is of principal typeNext we shall show that M is invertible at w0 if Im M P ge cPlowast P at w0 for some c gt 0 Then we

find as before from Lemma 46 that Ran M P(w0)perp Ker M P(w0) By choosing a base for Ker P(w0)

and completing it to a base of CN we may assume that

P(w0)=

(0 P12(w0)

0 P22(w0)

)where P22 is (N minus K )times (N minus K ) system K = Dim Ker P(w0) Now by multiplying P from the leftwith an orthogonal matrix E we may assume that P12(w0)= 0 In fact this only amounts to choosing anorthonormal base for Ran P(w0)

perp and completing to an orthonormal base for CN Observe that M P isunchanged if we replace M with M Eminus1 which is invertible if and only if M is Since Dim Ker P(w0)= Kwe obtain |P22(w0)| 6= 0 Letting

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

In fact (M P)12(w0) = M12(w0)P22(w0) = 0 since Ran M P(w0) = Ker M P(w0)perp We obtain that

M12(w0)= 0 and by assumption

Im M22 P22 ge cPlowast

22 P22 at w0

which gives |M22(w0)| 6= 0 Since P11 P21 and M12 vanish at w0 we find

Re V (M P)11(w0)= Re M11(w0)V P11(w0) gt c

which gives |M11(w0)| 6=0 Since M12(w0)=0 and |M22(w0)| 6=0 we obtain that M(w0) is invertible

Remark 48 The N times N system P isin Cinfin(T lowastRn) is quasisymmetrizable with respect to V if and onlyif there exists an invertible symmetrizer M such that Q = M P satisfies

Re〈(V Q)u u〉 ge cu2minus CQu

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

In fact by the CauchyndashSchwarz inequality we find

|〈(V M)Pu u〉| le εu2+ CεPu

2forall ε gt 0 forall u isin CN

Since M is invertible we also have that Pu sim= Qu

Definition 49 If the N times N system Q isin Cinfin(T lowastRn) satisfies (48)ndash(49) then Q is quasisymmetric withrespect to the real Cinfin vector field V

Proposition 410 Let P(w) isin Cinfin(T lowastRn) be an N times N quasisymmetrizable system then Plowast is qua-sisymmetrizable If A(w) and B(w) isin Cinfin(T lowastRn) are invertible N times N systems then B P A is quasisym-metrizable

Proof Clearly (48)ndash(49) are invariant under left multiplication of P with invertible systems E justreplace M with M Eminus1 Since we may write B P A = B(Alowast)minus1 Alowast P A it suffices to show that Elowast P E isquasisymmetrizable if E is invertible By Remark 48 there exists a symmetrizer M so that Q = M P isquasisymmetric that is satisfies (48)ndash(49) It then follows from Proposition 411 that

QE = ElowastQE = ElowastM(Elowast)minus1 Elowast P E

also satisfies (48) and (49) so Elowast P E is quasisymmetrizableFinally we shall prove that Plowast is quasisymmetrizable if P is Since Q = M P is quasisymmetric we

find from Proposition 411 that Qlowast= PlowastMlowast is quasisymmetric By multiplying with (Mlowast)minus1 from the

right we find from the first part of the proof that Plowast is quasisymmetrizable

Proposition 411 If Q isin Cinfin(T lowastRn) is quasisymmetric then Qlowast is quasisymmetric If E isin Cinfin(T lowastRn)

is invertible then ElowastQE are quasisymmetric

Proof First we note that (48) holds if and only if

Re〈(V Q)u u〉 ge cu2

forall u isin Ker Q (410)

for some cgt 0 In fact QlowastQ has a positive lower bound on the orthogonal complement Ker Qperp so that

u le CQu for u isin Ker Qperp

Thus if u = uprime+ uprimeprime with uprime

isin Ker Q and uprimeprimeisin Ker Qperp we find that Qu = Quprimeprime

Re〈(V Q)uprime uprimeprime〉 ge minusεuprime

2minus Cεuprimeprime

2ge minusεuprime

2minus C prime

forall ε gt 0

and Re〈(V Q)uprimeprime uprimeprime〉 ge minusCuprimeprime

ge minusC primeQu

2 By choosing ε small enough we obtain (48) by us-ing (410) on uprime

Next we note that Im Qlowast= minusIm Q and Re Qlowast

= Re Q so minusQlowast satisfies (49) and (410) with V re-placed by minusV and thus it is quasisymmetric Finally we shall show that QE = ElowastQE is quasisymmetricwhen E is invertible We obtain from (49) that

Im〈QE u u〉 = Im〈QEu Eu〉 ge 0 forall u isin CN

Next we shall show that QE satisfies (410) on Ker QE = Eminus1 Ker Q which will give (48) We findfrom Leibnizrsquo rule that V QE = (V Elowast)QE + Elowast(V Q)E + ElowastQ(V E) where (410) gives

Re〈Elowast(V Q)Eu u〉 ge cEu2ge cprime

u2 u isin Ker QE

since then Eu isin Ker Q Similarly we obtain that 〈(V Elowast)QEu u〉 = 0 when u isin Ker QE Now sinceIm QE ge 0 we find from Lemma 46 that

Ker Qlowast

E = Ker QE

346 NILS DENCKER

which gives 〈ElowastQ(V E)u u〉 = 〈Eminus1(V E)u Qlowast

E u〉 = 0 when u isin Ker QE = Ker Qlowast

E Thus QE satis-fies (410) so it is quasisymmetric

Example 412 Assume that P(w) isin Cinfin is an N times N system such that z isin6(P) (6ws(P)cap6infin(P))and that P(w)minusλ IdN is of principal type when |λminus z| 1 By Lemma 215 and Proposition 35 thereexists a Cinfin germ of eigenvalues λ(w) isin Cinfin for P so that Dim Ker(P(w)minus λ(w) IdN ) is constant near6z(P) By using the spectral projection as in the proof of Proposition 35 and making a base changeB(w) isin Cinfin we obtain

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

in a neighborhood of 6z(P) here |P22 minus λ(w) Id| 6= 0 We find from Proposition 35 that dλ 6= 0 whenλ = z so λminus z is of principal type Proposition 410 gives that P minus z IdN is quasisymmetrizable nearany w0 isin6z(P) if z isin part6(λ) In fact by Example 44 there exists q(w) isin Cinfin so that

|d Re q(λminus z)| 6= 0 (412)

Im q(λminus z)ge 0 (413)

and we get the normal form (42) for λ near 6z(P) = λ(w)= z One can then take V normal to6 = Re q(λminus z)= 0 at 6z(P) and use

M = Blowast

(q IdK 0

with M22(w)= (P22(w)minus z Id)minus1 for example Then

Q = M(P minus z IdN )= Blowast

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

If there are no closed semibicharacteristics of λminus z then we also find from Example 44 that P minus z IdN isquasisymmetrizable in a neighborhood of 6z(P) see the proof of Lemma 41 in [Dencker et al 2004]

where 0 le K (x) isin Cinfin When z gt 0 we find that P minus z IdN is quasisymmetric in a neighborhood of6z(P) with respect to the exterior normal 〈ξ partξ 〉 to 6z(P)=

|ξ |2 = z

For scalar symbols we find that 0 isin part6(p) if and only if p is quasisymmetrizable see Example 44But in the system case this needs not be the case according to the following example

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

which is quasisymmetrizable with respect to partw1 with symmetrizer

(0 11 0

In fact partw1 M P = Id2 and

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

so Im M P equiv 0 Since eigenvalues of P(w) are w2 plusmn

w21 minusw2

3 we find that 6(P)= C so 0 isin

6(P) isnot a boundary point of the eigenvalues

For quasisymmetrizable systems we have the following result

Theorem 415 Let the N times N system P(h) be given by (22) with principal symbol P isin Cinfin

b (TlowastRn)

Assume that z isin 6infin(P) and there exists a real valued function T (w) isin Cinfin such that P(w)minus z IdN isquasisymmetrizable with respect to the Hamilton vector field HT (w) in a neighborhood of 6z(P) Thenfor any K gt 0 we have

ζ isin C |ζ minus z|lt K h log(1h)cap Spec(P(h))= empty (415)

for 0lt h 1 and|(P(h)minus z)minus1

| le Ch 0lt h 1 (416)

If P is analytic in a tubular neighborhood of T lowastRn then there exists c0 gt 0 such thatζ isin C |ζ minus z|lt c0

cap Spec(P(h))= empty

Condition (416) means that λ isin3sc1 (P) which is the pseudospectrum of index 1 by Definition 227

The reason for the difference between (415) and (416) is that we make a change of norm in the proofthat is not uniform in h The conditions in Theorem 415 give some geometrical information on thebicharacteristic flow of the eigenvalues according to the following result

Remark 416 The conditions in Theorem 415 imply that the limit set at 6z(P) of the nontrivialsemibicharacteristics of the eigenvalues close to zero of Q = M(P minus z IdN ) is a union of compactcurves on which T is strictly monotone thus they cannot form closed orbits

In fact locally (w λ) isin1(P) 4(P) if and only if λ= λ(w) isin Cinfin by Lemma 215 Since P(w)minusλ IdN is of principal type by Proposition 47 we find that Dim Ker(P(w)minus λ(w) IdN ) is constant byProposition 35 Thus we obtain the normal form (414) as in Example 412 This shows that the Hamiltonvector field Hλ of an eigenvalue is determined by 〈d Qu u〉 with 0 6= u isin Ker(P minus ν IdN ) for ν close toz =λ(w) by the invariance property given by (32) Now 〈(HT Re Q)u u〉gt0 for 0 6=u isinKer(Pminusz IdN )and d〈Im Qu u〉 = 0 for u isin Ker M(P minus z IdN ) by (49) Thus by picking subsequences we find thatthe limits of nontrivial semibicharacteristics of eigenvalues λ of Q close to 0 give curves on which T isstrictly monotone Since z isin 6infin(P) these limit bicharacteristics are compact and cannot form closedorbits

P(x ξ)= |ξ |2 IdN +i K (x)

where 0 le K (x) isin Cinfin Then for z gt 0 we find that P minus z IdN is quasisymmetric in a neighborhood of6z(P) with respect to V = HT for T (x ξ) = minus〈ξ x〉 If K (x) isin Cinfin

b and 0 isin 6infin(K ) then we obtain

348 NILS DENCKER

from Proposition 220 Remark 221 Example 222 and Theorem 415 that

(Pw(x h D)minus z)minus1 le Ch 0lt h 1

since z isin6infin(P)

Proof of Theorem 415 We shall first consider the Cinfin

b case We may assume without loss of generalitythat z = 0 and we shall follow the proof of Theorem 13 in [Dencker et al 2004] By the conditions wefind from Definition 45 Remark 48 and (46) that there exists a function T (w) isin Cinfin

0 and a multiplierM(w) isin Cinfin

b (TlowastRn) so that Q = M P satisfies

Re HT Q ge c minus C Im Q (417)

Im Q ge c QlowastQ (418)

for some c gt 0 and then M is invertible by Proposition 47 In fact outside a neighborhood of 60(P)we have Plowast P ge c0 then we may choose M = i Plowast so that Q = i Plowast P and use a partition of unity to geta global multiplier Let

C1h le ε le C2h log 1h

where C1 1 will be chosen large Let T = Tw(x h D)

Q(h)= Mw(x h D)P(h)= Qw(x h D)+ O(h) (420)

Qε(h)= eεTh Q(h)eminusεTh= e

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

where adT Q(h)= [T (h) Q(h)] = O(h) By the assumption on ε and the boundedness of adT h we findthat the asymptotic expansion makes sense Since ε2

= O(h) we see that the symbol of Qε(h) is equal to

Qε = Q + iεT Q + O(h)

Since T is a scalar function we obtain

Im Qε = Im Q + εRe HT Q + O(h) (421)

Now to simplify notation we drop the parameter h in the operators Q(h) and P(h) and we shall use thesame letters for operators and the corresponding symbols Using (417) and (418) in (421) we obtainfor small enough ε that

Im Qε ge (1 minus Cε) Im Q + cεminus Ch ge cεminus Ch (422)

Since the symbol of 12i (Qε minus (Qε)

lowast) is equal to the expression (422) modulo O(h) the sharp Gardinginequality for systems in Proposition A5 gives

Im〈Qεu u〉 ge (cεminus C0h)u2geεc2

for h ε 1 By using the CauchyndashSchwarz inequality we obtainεc2

u le Qεu (423)

Since Q = M P the calculus givesQε = MεPε + O(h) (424)

where Pε = eminusεTh PeεTh and Mε = eminusεTh MeεTh= M + O(ε) is bounded and invertible for small

enough ε For h ε we obtain from (423)ndash(424) that

u leCε

Pεu (425)

so Pε is injective with closed range Now minusQlowast satisfies the conditions (43)ndash(44) with T replacedby minusT Thus we also obtain the estimate (423) for Qlowast

ε = Plowastε Mlowast

ε +O(h) Since Mlowastε is invertible for small

enough h we obtain the estimate (425) for Plowastε thus Pε is surjective Because the conjugation by eεTh

is uniformly bounded on L2 when ε le Ch we obtain the estimate (416) from (425)Now conjugation with eεTh is bounded in L2 (but not uniformly) also when (419) holds By taking C2

arbitrarily large in (419) we find from the estimate (425) for Pε and Plowastε that

0 K h log 1h

)cap Spec(P)= empty

for any K gt 0 when h gt 0 is sufficiently small

The analytic case We assume as before that z = 0 and

P(h)sim

sumjge0

h j Pwj (x h D) P0 = P

where the Pj are bounded and holomorphic in a tubular neighborhood of T lowastRn satisfy (23) andPwj (z h Dz) is defined by the formula (21) where we may change the integration to a suitable chosencontour instead of T lowastRn (see [Sjostrand 1982 Section 4]) As before we shall follow the proof ofTheorem 13 in [Dencker et al 2004] and use the theory of the weighted spaces H(3T ) developed in[Helffer and Sjostrand 1990] (see also [Martinez 2002])

The complexification T lowastCn of the symplectic manifold T lowastRn is equipped with a complex symplecticform ωC giving two natural real symplectic forms ImωC and ReωC We find that T lowastRn is Lagrangianwith respect to the first form and symplectic with respect to the second In general a submanifoldsatisfying these two conditions is called an IR-manifold

Assume that T isin Cinfin

0 (TlowastRn) then we may associate to it a natural family of IR-manifolds

3T = w+ iHT (w) w isin T lowastRn sub T lowastCn with isin R and || small

where as before we identify T (T lowastRn) with T lowastRn see [Dencker et al 2004 page 391] Since Im(ζdz)is closed on 3T we find that there exists a function G on 3T such that

dG = minusIm(ζdz)|3T

In fact we can write it down explicitly by parametrizing 3T by T lowastRn

G(z ζ )= minus〈ξ nablaξT (x ξ)〉 + T (x ξ) for (z ζ )= (x ξ)+ iHT (x ξ)

The associated spaces H(3T ) are going to be defined by using the FBI transform

T L2(Rn)rarr L2(T lowastRn)

350 NILS DENCKER

given by

T u(x ξ)= cnhminus3n4int

Rnei(〈xminusyξ〉+i |xminusy|

2)(2h)u(y) dy (426)

The FBI transform may be continued analytically to 3T so that T3T u isin Cinfin(3T ) Since 3T differsfrom T lowastRn on a compact set only we find that T3T u is square integrable on 3T The FBI transformcan of course also be defined on u isin L2(Rn) having values in CN and the spaces H(3T ) are definedby putting h dependent norms on L2(Rn)

u2H(3T )

|T3T u(z ζ )|2eminus2G(zζ )h(ω|3T )nn = T3T u

2L2(h)

Suppose that P1 and P2 are bounded and holomorphic N times N systems in a neighbourhood of T lowastRn

in T lowastCn and that u isin L2(RnCN ) Then we find for gt 0 small enough

〈Pw1 (x h D)u Pw2 (x h D)v〉H(3T )

= 〈(P1|3T )T3T u (P2|3T )T3T v〉L2(h) + O(h)uH(3T )vH(3T )

By taking P1 = P2 = P and u = v we obtain

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

as in the scalar case see [Helffer and Sjostrand 1990] or [Martinez 2002]By Remark 48 we may assume that M P = Q satisfies (48)ndash(49) with invertible M The analyticity

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

by Taylorrsquos formula thus

Im M(w)P(w+ iHT (w))= Im Q(w)+ Re M(w)HT P(w)+ O(2)

Since we have Re M HT P gt cminusC Im Q cgt 0 by (48) and Im Q ge 0 by (49) we obtain for sufficientlysmall gt 0 that

Im M(w)P(w+ iHT (w))ge (1 minus C) Im Q(w)+ c+ O(2)ge c2 (428)

which gives by the CauchyndashSchwarz inequality that P 3T u ge cprimeu Thus

Pminus1 3T le C (429)

Now recall that H(3T ) is equal to L2 as a space and that the norms are equivalent for every fixed h (butnot uniformly) Thus the spectrum of P(h) does not depend on whether the operator is realized on L2

or on H(3T ) We conclude from (427) and (429) that 0 has an h-independent neighbourhood whichis disjoint from the spectrum of P(h) when h is small enough

Summing up we have proved the following result

Proposition 418 Assume that P(h) is an N times N system on the form given by (22) with analytic princi-pal symbol P(w) and that there exists a real valued function T (w)isin Cinfin(T lowastRn) such that P(w)minusz IdN

is quasisymmetrizable with respect to HT in a neighborhood of 6z(P) Define the IR-manifold

3T = w+ iHT (w) w isin T lowastRn

for gt 0 small enough Then

P(h)minus z H(3T ) minusrarr H(3T )

has a bounded inverse for h small enough which gives

Spec(P(h))cap D(z δ)= empty 0lt h lt h0

for δ small enough

Remark 419 It is clear from the proof of Theorem 415 that in the analytic case it suffices that Pj isanalytic in a fixed complex neighborhood of 6z(P)b T lowastRn j ge 0

5 The subelliptic case

We shall investigate when we have an estimate of the resolvent which is better than the quasisymmetricestimate for example the subelliptic type of estimate

le Chminusmicro h rarr 0

with microlt 1 which we obtain in the scalar case under the bracket condition see Theorem 14 in [Denckeret al 2004]

Example 51 Consider the scalar operator pw = h Dt + i f w(t x h Dx) where 0 le f (t x ξ) isin Cinfin

b (t x) isin RtimesRn and 0 isin part6( f ) Then we obtain from Theorem 14 in [Dencker et al 2004] the estimate

hkk+1u le Cpwu h 1 forall u isin Cinfin

0 (51)

if 0 isin6infin( f ) and sumjlek

|partj

t f | 6= 0 (52)

These conditions are also necessary For example if | f (t)| le C |t |k then an easy computation givesh Dt u + i f uu le chkk+1 if u(t)= φ(thminus1k+1) with 0 6= φ(t) isin Cinfin

The following example shows that condition (52) is not sufficient for systems

Example 52 Let P = h Dt Id2 +i F(t) where

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

which gives that ⋂jle3

Ker F ( j)(0)= 0

But by taking u(t)=χ(t)(tminus1)t with 0 6=χ(t)isinCinfin

0 (R) we obtain F(t)u(t)equiv0 so we find Puule

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

Thus F(t)= t2 Blowast(t)5(t)B(t) where B(t) is invertible and 5(t) is a projection of rank one

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

Thus we find from the scalar case that h67u le CPu for h 1 see [Dencker et al 2004 Theo-

rem 14] Observe that this operator is element for element a higher order perturbation of the operatorof Example 52

Definition 54 Let 0 le F(t) isin Linfin

loc(R) be an N times N system then we define

δ(F)=

u=1〈F(t)u u〉 le δ

0lt δ le 1

which is well-defined almost everywhere and contains 60(F)= |F |minus1(0)

Observe that one can also use this definition in the scalar case then δ( f )= f minus1([0 δ]) for nonneg-ative functions f

Remark 55 Observe that if F ge 0 and E is invertible then we find that

δ(ElowastF E)subeCδ(F)

where C = Eminus1

Example 56 For the scalar symbols p(x ξ)= τ + i f (t x ξ) in Example 51 we find from PropositionA1 that (52) is equivalent to

|t f (t x ξ)le δ| = |δ( fxξ )| le Cδ1k 0lt δ 1 forall x ξ

where fxξ (t)= f (t x ξ)

Example 57 For the matrix F(t) in Example 53 we find from Remark 55 that |δ(F)| le Cδ16 andfor the matrix in Example 52 we find that |δ(F)| = infin

We also have examples when the semidefinite imaginary part vanishes of infinite order

Example 58 Let p(x ξ)= τ + i f (t x ξ) where 0 le f (t x ξ)le Ceminus1|t |σ σ gt 0 then we obtain that

|δ( fxξ )| le C0|log δ|minus1σ 0lt δ 1 forall x ξ

(We owe this example to Y Morimoto)

The following example shows that for subelliptic type of estimates it is not sufficient to have condi-tions only on the vanishing of the symbol we also need conditions on the semibicharacteristics of theeigenvalues

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

)+ i(t minusβx)2 Id2 (t x) isin R2

with α β isin R Then we see from the scalar case that P satisfies the estimate (51) with micro= 23 if andonly either α = 0 or α 6= 0 and β 6= plusmn1α

Definition 510 Let Q(w) isin Cinfin(T lowastRn) be an N times N system and let w0 isin 6 sub T lowastRn We say that Qsatisfies the approximation property on 6 near w0 if there exists a Q invariant Cinfin subbundle V of CN

over T lowastRn such that V(w0)= Ker QN (w0) and

Re〈Q(w)v v〉 = 0 v isin V(w) w isin6 (53)

near w0 That V is Q invariant means that Q(w)v isin V(w) for v isin V(w)

Here Ker QN (w0) is the space of the generalized eigenvectors corresponding to the zero eigenvalueThe symbol of the system in Example 59 satisfies the approximation property on 6 = τ = 0 if andonly if α = 0

Let Q = Q|V then since Im i Q = Re Q we obtain from Lemma 46 that Ran Qperp Ker Q on 6 ThusKer QN

= Ker Q on6 and since Ker QN (w0)= V(w0)we find that Ker QN (w0)= V(w0)= Ker Q(w0)It follows from Example 513 that Ker Q sube V near w0

Remark 511 Assume that Q satisfies the approximation property on the Cinfin hypersurface 6 and isquasisymmetric with respect to V isin T6 Then the limits of the nontrivial semibicharacteristics of theeigenvalues of Q close to zero coincide with the bicharacteristics of 6

In fact the approximation property in Definition 510 and Example 513 give that 〈Re Qu u〉 = 0 foru isin Ker Q sube V on 6 Since Im Q ge 0 we find that

〈d Qu u〉 = 0 forall u isin Ker Q on T6 (54)

By Remark 416 the limits of the nontrivial semibicharacteristics of the eigenvalues close to zero of Qare curves with tangents determined by 〈d Qu u〉 for u isin Ker Q Since V Re Q 6= 0 on Ker Q we findfrom (54) that the limit curves coincide with the bicharacteristics of 6 which are the flow-outs of theHamilton vector field

Example 512 Observe that Definition 510 is empty if Dim Ker QN (w0)= 0 If Dim Ker QN (w0) gt 0then there exists ε gt 0 and a neighborhood ω to w0 so that

5(w)=1

int|z|=ε

(z IdN minusQ(w))minus1 dz isin Cinfin(ω) (55)

354 NILS DENCKER

is the spectral projection on the (generalized) eigenvectors with eigenvalues having absolute value lessthan ε Then Ran5 is a Q invariant bundle over ω so that Ran5(w0) = Ker QN (w0) Condition (53)with V = Ran5 means that 5lowast Re Q5 equiv 0 in ω When Im Q(w0) ge 0 we find that 5lowastQ5(w0) = 0then Q satisfies the approximation property on 6 near w0 with V = Ran5 if and only if

d(5lowast(Re Q)5)|T6 equiv 0 near w0

Example 513 If Q satisfies the approximation property on 6 then by choosing an orthonormal basefor V and extending it to an orthonormal base for CN we obtain the system on the form

(Q11 Q12

where Q11 is K times K system such that QN11(w0) = 0 Re Q11 = 0 on 6 and |Q22| 6= 0 By multiplying

from the left with (IdK minusQ12 Qminus1

220 IdNminusK

)we obtain that Q12 equiv 0 without changing Q11 or Q22

In fact the eigenvalues of Q are then eigenvalues of either Q11 or Q22 Since V(w0) are the (general-ized) eigenvectors corresponding to the zero eigenvalue of Q(w0)we find that all eigenvalues of Q22(w0)

are nonvanishing thus Q22 is invertible near w0

Remark 514 If Q satisfies the approximation property on6 nearw0 then it satisfies the approximationproperty on 6 near w1 for w1 sufficiently close to w0

In fact let Q11 be the restriction of Q to V as in Example 513 then since Re Q11 = Im i Q11 = 0 on6we find from Lemma 46 that Ran Q11perp Ker Q11 and Ker Q11 = Ker QN

11 on 6 Since Q22 is invertiblein (56) we find that Ker Q sube V Thus by using the spectral projection (55) of Q11 near w1 isin 6 forsmall enough ε we obtain an invariant subbundle V sube V so that V(w1)= Ker Q11(w1)= Ker QN (w1)

If Q isin Cinfin satisfies the approximation property and QE = ElowastQE with invertible E isin Cinfin then itfollows from the proof of Proposition 518 below that there exist invertible A B isin Cinfin such that AQE

and QlowastB satisfy the approximation property

Definition 515 Let P isin Cinfin(T lowastRn) be an N times N system and let φ(r) be a positive nondecreasingfunction on R+ We say that P is of subelliptic type φ if for anyw0 isin60(P) there exists a neighborhoodωofw0 a Cinfin hypersurface6 3w0 a real Cinfin vector field V isin T6 and an invertible symmetrizer M isin Cinfin

so that Q = M P is quasisymmetric with respect to V in ω and satisfies the approximation property on6 capω Also for every bicharacteristic γ of 6 the arc length∣∣γ capδ(Im Q)capω

∣∣ le Cφ(δ) 0lt δ 1 (57)

We say that z is of subelliptic type φ for P isin Cinfin if P minus z IdN is of subelliptic type φ If φ(δ)= δmicro thenwe say that the system is of finite type of order microge 0 which generalizes the definition of finite type forscalar operators in [Dencker et al 2004]

Recall that the bicharacteristics of a hypersurface in T lowastX are the flow-outs of the Hamilton vectorfield of 6 Of course if P is elliptic then by choosing M = i Pminus1 we obtain Q = i IdN so P is trivially

of subelliptic type If P is of subelliptic type then it is quasisymmetrizable by definition and thus ofprincipal type

Remark 516 Observe that we may assume that

Im〈Qu u〉 ge cQu2

forall u isin CN (58)

in Definition 515

In fact by adding iPlowast to M we obtain (58) for large enough by (46) and this only increasesIm Q

Since Q is in Cinfin the estimate (57) cannot be satisfied for any φ(δ) δ (unless Q is elliptic) andit is trivially satisfied with φ equiv 1 thus we shall only consider cδ le φ(δ) 1 (or finite type of order0 lt micro le 1) Actually for Cinfin symbols of finite type the only relevant values in (57) are micro = 1k foreven k gt 0 see Proposition A2 in the Appendix

Actually the condition that φ is nondecreasing is unnecessary since the left-hand side in (57) is non-decreasing (and upper semicontinuous) in δ we can replace φ(δ) by infεgtδ φ(ε) to make it nondecreasing(and upper semicontinuous)

Example 517 Assume that Q is quasisymmetric with respect to the real vector field V satisfying (57)and the approximation property on 6 Then by choosing an orthonormal base as in Example 513 weobtain the system on the form

(Q11 Q12

)where Q11 is K times K system such that QN

11(w0) = 0 Re Q11 = 0 on 6 and |Q22| 6= 0 Since Q isquasisymmetric with respect to V we also obtain that Q11(w0)= 0 Re V Q11 gt 0 Im Q j j ge 0 for j = 12 In fact then Lemma 46 gives that Im Qperp Ker Q so Ker QN

= Ker Q Since Q satisfies (57) andδ(Im Q11) sube δ(Im Q) we find that Q11 satisfies (57) By multiplying from the left as in Example513 we obtain that Q12 equiv 0 without changing Q11 or Q22

Proposition 518 If the NtimesN system P(w)isinCinfin(T lowastRn) is of subelliptic type φ then Plowast is of subelliptictype φ If A(w) and B(w)isin Cinfin(T lowastRn) are invertible N times N systems then AP B is of subelliptic type φ

Proof Let M be the symmetrizer in Definition 515 so that Q = M P is quasisymmetric with respect toV By choosing a suitable base and changing the symmetrizer as in Example 517 we may write

(Q11 00 Q22

where Q11 is K times K system such that Q11(w0) = 0 V Re Q11 gt 0 Re Q11 = 0 on 6 and that Q22 isinvertible We also have Im Q ge 0 and that Q satisfies (57) Let V1 = u isin CN

u j = 0 for j gt K andV2 = u isin CN

u j = 0 for j le K these are Q invariant bundles such that V1 oplus V2 = CN First we are going to show that P = AP B is of subelliptic type By taking M = Bminus1 M Aminus1 we find

M P = Q = Bminus1 Q B

356 NILS DENCKER

and it is clear that Bminus1V j are Q invariant bundles j = 1 2 By choosing bases in Bminus1V j for j = 1 2we obtain a base for CN in which Q has a block form

(Q11 00 Q22

)Here Q j j Bminus1V j 7rarr Bminus1V j is given by Q j j = Bminus1

j Q j j B j with

B j Bminus1V j 3 u 7rarr Bu isin V j j = 1 2

By multiplying Q from the left with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

It is clear that Im Q ge 0 Q11(w0) = 0 Re Q11 = 0 on 6 |Q22| 6= 0 and V Re Q11 gt 0 by Proposition411 Finally we obtain from Remark 55 that

δ(Im Q)subeCδ(Im Q)

for some C gt 0 which proves that P = AP B is of subelliptic type Observe that Q = AQ B whereQ B = BlowastQ B and A = BBminus1(Blowast)minus1

To show that Plowast also is of subelliptic type we may assume as before that Q = M P is on the form (59)with Q11(w0) = 0 V Re Q11 gt 0 Re Q11 = 0 on 6 Q22 is invertible Im Q ge 0 and Q satisfies (57)Then we find that

minusPlowastMlowast= minusQlowast

(minusQlowast

11 00 minusQlowast

)satisfies the same conditions with respect to minusV so it is of subelliptic type with multiplier IdN By thefirst part of the proof we obtain that Plowast is of subelliptic type

Example 519 In the scalar case p isin Cinfin(T lowastRn) is quasisymmetrizable with respect to Ht = partτ near w0

if and only if

p(t x τ ξ)= q(t x τ ξ)(τ + i f (t x ξ)) near w0 (510)

with f ge 0 and q 6= 0 see Example 44 If 0 isin 6infin(p) we find by taking qminus1 as symmetrizer that pin (510) is of finite type of order micro if and only if micro= 1k for an even k such thatsum

|partkt f |gt 0

by Proposition A1 In fact the approximation property is trivial since f is real Thus we obtain the casein [Dencker et al 2004 Theorem 14] see Example 51

Theorem 520 Assume that the N times N system P(h) is given by the expansion (22) with principalsymbol P isin Cinfin

b (TlowastRn) Assume that z isin 6(P) 6infin(P) is of subelliptic type φ for P where φ gt 0 is

nondecreasing on R+ Then there exists h0 gt 0 so that

(P(h)minus z IdN )minus1

le Cψ(h) 0lt h le h0 (511)

where ψ(h)= δ is the inverse to h = δφ(δ) It follows that there exists c0 gt 0 such that

w |wminus z| le c0ψ(h) cap σ(P(h))= empty 0lt h le h0

Theorem 520 will be proved in Section 6 Observe that if φ(δ)rarr c gt 0 as δrarr 0 then ψ(h)= O(h)and Theorem 520 follows from Theorem 415 Thus we shall assume that φ(δ)rarr 0 as δ rarr 0 then wefind that h = δφ(δ) = o(δ) so ψ(h) h when h rarr 0 In the finite type case φ(δ) = δmicro we find thatδφ(δ) = δ1+micro and ψ(h) = h1micro+1 When micro = 1k we find that 1 +micro = (k + 1)k and ψ(h) = hkk+1Thus Theorem 520 generalizes Theorem 14 in [Dencker et al 2004] by Example 519 Condition (511)with ψ(h)= h1micro+1 means that λ isin3sc

1micro+1(P) which is the pseudospectrum of index 1micro+ 1

Example 521 Assume that P(w) isin Cinfin is N times N and z isin6(P) (6ws(P)cup6infin(P)) Then 6micro(P)=λ(w)= micro for micro close to z where λ isin Cinfin is a germ of eigenvalues for P at 6z(P) see Lemma 215If z isin part6(λ) we find from Example 412 that P(w)minus z IdN is quasisymmetrizable near w0 isin 6z(P)if P(w)minus λ IdN is of principal type when |λminus z| 1 Then P is on the form (411) and there existsq(w) isin Cinfin so that (412)ndash(413) hold near 6z(P) We can then choose the multiplier M so that Q ison the form (414) By taking 6 = Re q(λminus z)= 0 we obtain that P minus z IdN is of subelliptic type φif (57) is satisfied for Im q(λminus z) In fact by the invariance we find that the approximation property istrivially satisfied since Re qλequiv 0 on 6

Example 522 LetP(x ξ)= |ξ |2 IdN +i K (x) (x ξ) isin T lowastRn

where K (x) isin Cinfin(Rn) is symmetric as in Example 312 We find that P minus z IdN is of finite type oforder 12 when z = iλ for almost all λ isin 6(K ) (6ws(K ) cup6infin(K )) by Example 521 In fact thenz isin6(P)(6ws(P)cap6infin(P)) and the Cinfin germ of eigenvalues for P near6z(P) is λ(x ξ)=|ξ |2+iκ(x)where κ(x) is a Cinfin germ of eigenvalues for K (x) near 6λ(K ) = κ(x)= λ For almost all values λwe have dκ(x) 6= 0 on 6λ(K ) By taking q = i we obtain for such values that (57) is satisfied forIm i(λ(w) minus iλ) = |ξ |2 with φ(δ) = δ12 since Re i(λ(w) minus iλ) = λ minus κ(x) = 0 on 6 = 6λ(K )If K (x) isin Cinfin

b and 0 isin 6infin(K ) then we may use Theorem 520 Proposition 220 Remark 221 andExample 222 to obtain the estimate

(Pw(x h D)minus z IdN )minus1

le Chminus23 0lt h 1

on the resolvent

Example 523 LetP(t x τ ξ)= τM(t x ξ)+ i F(t x ξ) isin Cinfin

where M ge c0 gt 0 and F ge 0 satisfies∣∣∣t inf|u|=1

〈F(t x ξ)u u〉 le δ∣∣∣ le Cφ(δ) forall x ξ (512)

358 NILS DENCKER

Then P is quasisymmetrizable with respect to partτ with symmetrizer IdN When τ = 0 we obtain thatRe P = 0 so by taking V = Ran5 for the spectral projection 5 given by (55) for F we find that Psatisfies the approximation property with respect to 6 = τ = 0 Since δ(Im P) = δ(F) we findfrom (512) that P is of subelliptic type φ Observe that if 0 isin 6infin(F) we obtain from Proposition A2that (512) is satisfied for φ(δ)= δmicro if and only if microle 1k for an even k ge 0 so thatsum

|partj

t 〈F(t x ξ)u(t) u(t)〉|gt 0 forall t x ξ

for any 0 6= u(t) isin Cinfin(R)

6 Proof of Theorem 520

By subtracting z IdN we may assume z = 0 Let w0 isin60(P) then by Definition 515 and Remark 516there exist a Cinfin hypersurface 6 and a real Cinfin vector field V isin T6 an invertible symmetrizer M isin Cinfin

so that Q = M P satisfies (57) the approximation property on 6 and

V Re Q ge c minus C Im Q (61)

in a neighborhood ω of w0 here c gt 0Since (61) is stable under small perturbations in V we can replace V with Ht for some real t isin Cinfin

after shrinkingω By solving the initial value problem Htτ equivminus1 τ |6=0 and completing to a symplecticCinfin coordinate system (t τ x ξ) we obtain that 6 = τ = 0 in a neighborhood of w0 = (0 0 w0) Weobtain from Definition 515 that

Re〈Qu u〉 = 0 when τ = 0 and u isin V (63)

near w0 Here V is a Q invariant Cinfin subbundle of CN such that V(w0)= Ker QN (w0)= Ker Q(w0) byLemma 46 By condition (57) we have that∣∣δ(Im Qw)cap |t |lt c

∣∣ le Cφ(δ) 0lt δ 1 (64)

when |wminusw0|lt c here Qw(t)= Q(t 0 w) Since these are all local conditions we may assume thatM and Q isin Cinfin

b We shall obtain Theorem 520 from the following estimate

Proposition 61 Assume that Q isin Cinfin

b (TlowastRn) is an N times N system satisfying (61)ndash(64) in a neighbor-

hood of w0 = (0 0 w0) with V = partτ and nondecreasing φ(δ) rarr 0 as δ rarr 0 Then there exist h0 gt 0and R isin Cinfin

b (TlowastRn) so that w0 isin supp R and

ψ(h)u le C(Qw(t x h Dtx)u +Rw(t x h Dtx)u + hu) 0lt h le h0 (65)

for any u isin Cinfin

0 (RnCN ) Here ψ(h)= δ h is the inverse to h = δφ(δ)

Let ω be a neighborhood of w0 such that supp R capω= empty where R is given by Proposition 61 Takeϕ isin Cinfin

0 (ω) such that 0 le ϕ le 1 and ϕ = 1 in a neighborhood of w0 By substituting ϕw(t x h Dtx)uin (65) we obtain from the calculus that for any N we have

ψ(h)ϕw(t x h Dtx)u le CN (Qw(t x h Dtx)ϕw(t x h Dtx)u + hN

u) forall u isin Cinfin

0 (66)

for small enough h since Rϕ equiv 0 Now the commutator

[Qw(t x h Dtx) ϕw(t x h Dtx)]u le Chu u isin Cinfin

Since Q = M P the calculus gives

Qw(t x h Dtx)u le Mw(t x h Dtx)P(h)u + Chu le C prime(P(h)u + hu) u isin Cinfin

0 (67)

The estimates (66)ndash(67) give

ψ(h)ϕw(t x h Dtx)u le C(P(h)u + hu) (68)

Since 0 isin6infin(P) we obtain by using the Borel Theorem finitely many functions φ j isin Cinfin

0 j = 1 N such that 0leφ j le1

sumj φ j =1 on60(P) and the estimate (68) holds with φ=φ j Let φ0 =1minus

sumjge1 φ j

then since 0 isin6infin(P) we find that Pminus1 le C on suppφ0 Thus φ0 = φ0 Pminus1 P and the calculus gives

φw0 (t x h Dtx)u le C(P(h)u + hu) u isin Cinfin

By summing up we obtain

ψ(h)u le C(P(h)u + hu) u isin Cinfin

0 (69)

Since h = δφ(δ) δ we find ψ(h)= δ h when h rarr 0 Thus we find for small enough h that the lastterm in the right hand side of (69) can be cancelled by changing the constant then P(h) is injective withclosed range Since Plowast(h) also is of subelliptic type φ by Proposition 518 we obtain the estimate (69)for Plowast(h) Thus Plowast(h) is injective making P(h) is surjective which together with (69) gives Theorem520

Proof of Proposition 61 First we shall prepare the symbol Q locally near w0 =(0 0 w0) Since Im Q ge0we obtain from Lemma 46 that Ran Q(w0)perp Ker Q(w0) which gives Ker QN (w0) = Ker Q(w0) LetDim Ker Q(w0)= K then by choosing an orthonormal base and multiplying from the left as in Example517 we may assume that

(Q11 00 Q22

)where Q11 is K times K matrix Q11(w0) = 0 and |Q22(w0)| 6= 0 Also we find that Q11 satisfies theconditions (61)ndash(64) with V = CK near w0

Now it suffices to prove the estimate with Q replaced by Q11 In fact by using the ellipticity of Q22

at w0 we find

uprimeprime le C(Qw

22uprimeprime +Rw1 uprimeprime

+ huprimeprime uprimeprime

isin Cinfin

0 (RnCNminusK )

where u = (uprime uprimeprime) and w0 isin supp R1 Thus if we have the estimate (65) for Qw11 with R = R2 then

since ψ(h) is bounded we obtain the estimate for Qw

ψ(h)u le C0(Qw11uprime

+Qw22uprimeprime

+Rwu + hu)le C1(Qwu +Rwu + hu)

where w0 isin supp R R = (R1 R2)

360 NILS DENCKER

Thus in the following we may assume that Q = Q11 is K times K system satisfying the conditions (61)ndash(64) with V = CK near w0 Since partτ Re Q gt 0 at w0 by (61) we find from the matrix version of theMalgrange Preparation Theorem in [Dencker 1993 Theorem 43] that

Q(t τ w)= E(t τ w)(τ Id +K0(t w)) near w0

where E K0 isin Cinfin and Re E gt 0 at w0 By taking M(t w)= E(t 0 w) we find Re M gt 0 and

Q(t τ w)= E0(t τ w)(τM(t w)+ i K (t w))= E0(t τ w)Q0(t τ w)

where E0(t 0 w)equiv Id Thus we find that Q0 satisfies (62) (63) and (64) when τ = 0 near w0 SinceK (0 w0)= 0 we obtain that Im K equiv 0 and K ge cK 2

ge 0 near (0 w0) We have Re M gt 0 and

|〈Im Mu u〉| le C〈K u u〉12

u near (0 w0) (610)

In fact we have0 le Im Q le K + τ(Im M + Re(E1K ))+ Cτ 2

where E1(t w)= partτ E(t 0 w) Lemma A7 gives

|〈Im Mu u〉 + Re〈E1K u u〉| le C〈K u u〉12

and since K 2le C K we obtain

|Re〈E1K u u〉| le CK uu le C0〈K u u〉12

which gives (610) Now by cutting off when |τ | ge c gt 0 we obtain that

Qw= Ew0 Qw

0 + Rw0 + h Rw1

where R j isin Cinfin

b and w0 isin supp R0 Thus it suffices to prove the estimate (65) for Qw0 We may now

reduce to the case when Re M equiv Id In fact

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

where M0 = (Re M)12 is invertible Mlowast

1 = M1 and K1 = Mminus10 K Mminus1

0 ge 0 By changing M1 and K1 andmaking K1 gt 0 outside a neighborhood of (0 w0) we may assume that M1 K1 isin Cinfin

b and i K1 satis-fies (64) for all c gt 0 and any w by the invariance given by Remark 55 Observe that condition (610)also is invariant under the mapping Q0 7rarr ElowastQ0 E

We shall use the symbol classes f isin S(m g) defined by

|partν1 partνk f | le Ckmkprod

g(ν j )12

forall ν1 νk forall k

for constant weight m and metric g and Op S(m g) the corresponding Weyl operators f w We shallneed the following estimate for the model operator Qw

Proposition 62 Assume that

Q = Mw(t x h Dx)h Dt + i Kw(t x h Dx)

where M(t w) and 0 le K (t w) isin Linfin(RCinfin

b (TlowastRn)) are N times N system such that Re M equiv Id Im M

satisfies (610) and i K satisfies (64) for all w and c gt 0 with non-decreasing 0 lt φ(δ) rarr 0 asδ rarr 0 Then there exists a real valued B(t w) isin Linfin(R S(1 H |dw|

2h)) such that h B(t w)ψ(h) isin

Lip(R S(1 H |dw|2h)) and

ψ(h)u2le Im〈Qu Bw(t x h Dx)u〉 + Ch2

Dt u2 0lt h 1 (611)

0 (Rn+1CN ) Here the bounds on B(t w) are uniform ψ(h) = δ h is the inverse to

h = δφ(δ) so 0lt H =radic

hψ(h) 1 as h rarr 0

Observe that H 2= hψ(h) = φ(ψ(h)) rarr 0 and hH =

radicψ(h)h ψ(h) rarr 0 as h rarr 0 since

0lt φ(δ) is non-decreasingTo prove Proposition 61 we shall cut off where |τ | ≷ ε

radicψh Take χ0(r) isin Cinfin

0 (R) such that0 le χ0 le 1 χ0(r) = 1 when |r | le 1 and |r | le 2 in suppχ0 Then 1 minus χ0 = χ1 where 0 le χ1 le 1 issupported where |r | ge 1 Let φ jε(r) = χ j (hrε

radicψ) j = 0 1 for ε gt 0 then φ0ε is supported where

|r | le 2εradicψh and φ1ε is supported where |r | ge ε

radicψh We have that φ jε(τ )isin S(1 h2dτ 2ψ) j = 0

1 and u = φ0ε(Dt)u +φ1ε(Dt)u where we shall estimate each term separately Observe that we shalluse the ordinary quantization and not the semiclassical for these operators

To estimate the first term we substitute φ0ε(Dt)u in (611) We find that

ψ(h)φ0ε(Dt)u2le Im〈Qu φ0ε(Dt)Bw(t x h Dx)φ0ε(Dt)u〉

+ Im〈[Q φ0ε(Dt)]u Bw(t x h Dx)φ0ε(Dt)u〉 + 4Cε2ψu2 (612)

In fact hDtφ0ε(Dt)u le 2εradicψu since it is a Fourier multiplier and |hτφ0ε(τ )| le 2ε

radicψ Next we

shall estimate the commutator term Since Re Q = h Dt Id minushpartt Im Mw2 and Im Q = h Im MwDt +

Kw+ hpartt Im Mw2i we find that [Re Q φ0ε(Dt)] isin Op S(hG) and

[Q φ0ε(Dt)] = i[Im Q φ0ε(Dt)] = i[Kw φ0ε(Dt)] = minushpartt Kwφ2ε(Dt)εradicψ

is a symmetric operator modulo Op S(hG) where G = dt2+h2dτ 2ψ+|dx |

|dξ |2 and φ2ε(τ )=

χ prime

0(hτεradicψ) In fact we have that h2ψ(h)le Ch h[partt Im Mw φ0ε(Dt)] and [Im Mw φ0ε(Dt)]h Dt isin

Op S(hG) since |τ | le εradicψh in suppφ0ε(τ ) Thus we find that

minus 2i Im(φ0ε(Dt)Bw[Q φ0ε(Dt)]

)= 2ihεminus1ψminus12 Im

(φ0ε(Dt)Bwpartt Kwφ2ε(Dt)

)= hεminus1ψminus12

(φ0ε(Dt)Bw[partt Kw φ2ε(Dt)] +φ0ε(Dt)[Bw φ2ε(Dt)]partt Kw

+φ2ε(Dt)[φ0ε(Dt) Bw]partt Kw+φ2ε(Dt)Bw[φ0ε(Dt) partt Kw

)(613)

modulo Op S(hG) As before the calculus gives that [φ jε(Dt) partt Kw] isin Op S(hψminus12G) for any

j Since t rarr h Bwψ isin Lip(ROp S(1G)) uniformly and φ jε(τ ) = χ j (hτεradicψ) with χ prime

j isin Cinfin

0 (R)Lemma A4 with κ = ε

radicψh gives that∥∥[φ jε(Dt) Bw]

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

uniformly If we combine the estimates above we can estimate the commutator term

| Im〈[Q φ0ε(Dt)]u Bw(t x h Dx)φ0ε(Dt)u〉| le Chu2 ψ(h)u

2 h 1 (614)

which together with (612) will give the estimate for the first term for small enough ε and hWe also have to estimate φ1ε(Dt)u then we shall use that Q is elliptic when |τ | 6= 0 We have

φ1ε(Dt)u2= 〈χw(Dt)u u〉

where χ(τ) = φ21ε(τ ) isin S(1 h2dτ 2ψ) is real with support where |τ | ge ε

radicψh Thus we may write

χ(Dt)=(Dt)h Dt where (τ)=χ(τ)hτ isin S(ψminus12 h2dτ 2ψ) by Leibnizrsquo rule since |τ |minus1le hε

radicψ

in supp Now h Dt Id = Re Q + hpartt Im Mw2 so we find

〈χ(Dt)u u〉 = Re〈(Dt)Qu u〉 +h2

Re〈(Dt)(partt Im Mw)u u〉 + Im〈(Dt) Im Qu u〉

where |h Re〈(Dt)(partt Im Mw)u u〉| le Chu2ε

radicψ and

| Re〈(Dt)Qu u〉| le Qu(Dt)u le Quuεradicψ

since (Dt) is a Fourier multiplier and |(τ)| le 1εradicψ We have that

Im Q = Kw(t x h Dx)+ h Dt Im Mw(t x h Dx)minush2ipartt Im Mw(t x h Dx)

where Im Mw(t x h Dx) and Kw(t x h Dx)isinOp S(1G) are symmetric Since =χhτ isin S(ψminus12G)

is real we find that

Im((Dt) Im Q)= Im (Dt)Kw+ Imχ(Dt) Im Mw

([(Dt) Kw(t x h Dx)] + [χ(Dt) Im Mw(t x h Dx)]

)modulo terms in Op S(h

radicψG)sube Op S(hψG) Here the calculus gives

[(Dt) Kw(t x h Dx)] isin Op S(hψG)

and similarly we have that

[χ(Dt) Im Mw(t x h Dx)] isin Op S(hradicψG)sube Op S(hψG)

which gives that |Im〈(Dt) Im Qu u〉| le Chu2ψ In fact since the metric G is constant it is uni-

formly σ temperate for all h gt 0 We obtain that

ψ(h)φ1ε(Dt)u2le Cε(

radicψQuu + hu

which together with (612) and (614) gives the estimate (65) for small enough ε and h since hψ(h)rarr0as h rarr 0

Proof of Proposition 62 We shall do a second microlocalization in w = (x ξ) By making a linearsymplectic change of coordinates (x ξ) 7rarr (h12x hminus12ξ) we see that Q(t τ x hξ) is changed into

Q(t τ h12w) isin S(1 dt2+ dτ 2

+ h|dw|2) when |τ | le c

In these coordinates we find B(h12w) isin S(1G) G = H |dw|2 if B(w) isin S(1 H |dw|

2h) In thefollowing we shall use ordinary Weyl quantization in the w variables

We shall follow an approach similar to the one of [Dencker et al 2004 Section 5] To localize theestimate we take φ j (w) j ψ j (w) j isin S(1G) with values in `2 such that 0 le φ j 0 le ψ j

sumj φ

2j equiv 1

and φ jψ j = φ j for all j We may also assume that ψ j is supported in a G neighborhood of w j This canbe done uniformly in H by taking φ j (w)=8 j (H 12w) and ψ j (w)=9 j (H 12w) with 8 j (w) j and9 j (w) j isin S(1 |dw|

2) Sincesumφ2

j = 1 and G = H |dw|2 the calculus givessum

φwj (x Dx)u2minus C H 2

u2le u

φwj (x Dx)u2+ C H 2

for u isin Cinfin

0 (Rn) thus for small enough H we findsum

φwj (x Dx)u2le 2u

φwj (x Dx)u2 for u isin Cinfin

0 (Rn) (615)

Observe that since φ j has values in `2 we find that φwj Rwj j isin Op S(H νG) also has values in `2 ifR j isin S(H νG) uniformly Such terms will be summablesum

rwj u2le C H 2ν

u2 (616)

for r j j isin S(H νG) with values in `2 see [Hormander 1983ndash1985 Volume III page 169] Now wefix j and let

Q j (t τ )= Q(t τ h12w j )= M j (t)τ + i K j (t)

where M j (t) = M(t h12w j ) and K j (t) = K (t h12w j ) isin Linfin(R) Since K (t w) ge 0 we find fromLemma A7 and (610) that

|〈Im M j (t)u u〉| + |〈dwK (t h12w j )u u〉| le C〈K j (t)u u〉12

u forall u isin CNforall t (617)

and condition (64) means that ∣∣∣t inf|u|=1

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

We shall prove an estimate for the corresponding one-dimensional operator

Q j (t h Dt)= M j (t)h Dt + i K j (t)

by using the following result

Lemma 63 Assume thatQ(t h Dt)= M(t)h Dt + i K (t)

where M(t) and 0 le K (t) are N times N systems which are uniformly bounded in Linfin(R) such thatRe M equiv Id Im M satisfies (610) for almost all t and i K satisfies (64) for any cgt 0 with non-decreasingφ(δ) rarr 0 as δ rarr 0 Then there exists a uniformly bounded real B(t) isin Linfin(R) so that h B(t)ψ(h) isin

Lip(R) uniformly and

ψ(h)u2+ 〈K u u〉 le Im〈Qu Bu〉 + Ch2

Dt u2 0lt h 1 (619)

364 NILS DENCKER

0 (RCN ) Here ψ(h)= δ h is the inverse to h = δφ(δ)

Proof Let 0 le8h(t)le 1 be the characteristic function of the set δ(K ) with δ =ψ(h) Since δ =ψ(h)is the inverse of h = δφ(δ) we find that φ(ψ(h))= hδ = hψ(h) Thus we obtain from (618) thatint

8h(t) dt = |δ(K )| le Chψ(h)

Letting

E(t)= exp(ψ(h)

08h(s) ds

we find that E and Eminus1isin Linfin(R)cap C0(R) uniformly and E prime

= ψ(h)hminus18h E in Dprime(R) We have

E(t)Q(t h Dt)Eminus1(t)= Q(t h Dt)+ E(t)h[M(t)Dt Eminus1(t)] IdN

= Q(t h Dt)+ iψ(h)8h(t) IdN minusψ(h)8h(t) Im M(t) (620)

since (Eminus1)prime = minusE primeEminus2 In the following we let

F(t)= K (t)+ψ(h) IdN ge ψ(h) IdN

By definition we have 8h(t) lt 1 HrArr K (t)ge ψ(h) IdN so

K (t)+ψ(h)8h(t) IdN ge12 F(t)

Thus by taking the inner product in L2(R) we find from (620) that

Im〈E(t)Q(t h Dt)Eminus1(t)u u〉 ge12〈F(t)u u〉 + 〈Im M(t)h Dt u u〉 minus chu

2 u isin Cinfin

0 (RCN )

since Im Q(t h Dt) = K (t)+ Im M(t)h Dt +h2i partt Im M(t) Now we may use (610) to estimate for any

ε gt 0|〈Im Mh Dt u u〉| le ε〈K u u〉 + Cε(h2

Dt u2+ hu

2) forall u isin Cinfin

0 (RCN ) (621)

In fact u = χ0(h Dt)u +χ1(h Dt)u where χ0(r) isin Cinfin

0 (R) and |r | ge 1 in suppχ1 We obtain from (610)for any ε gt 0 that

|〈Im M(t)χ0(hτ)hτu u〉| le C〈K (t)u u〉12

|χ0(hτ)hτ |u le ε〈K (t)u u〉 + Cεχ0(hτ)hτu2

so by using Gardings inequality in Proposition A5 on

εK (t)+ Cεχ20 (h Dt)h2 D2

t plusmn Im M(t)χ0(h Dt)h Dt

we obtain

|〈Im M(t)χ0(h Dt)h Dt u u〉| le ε〈K (t)u u〉 + Cεh2Dt u

2+ C0hu

2forall u isin Cinfin

0 (RCN )

since χ0(h Dt)h Dt u le Ch Dt u The other term is easier to estimate

|〈Im M(t)χ1(h Dt)h Dt u u〉| le Ch Dt uχ1(h Dt)u le C1h2Dt u

since |χ1(hτ)| le C |hτ | By taking ε = 16 in (621) we obtain

〈F(t)u u〉 le 3 Im〈E(t)Q(t h Dt)Eminus1(t)u u〉 + C(h2Dt u

Now h Dt Eu = Eh Dt u minus iψ(h)8h Eu so we find by substituting E(t)u that

ψ(h)E(t)u2+ 〈K E(t)u E(t)u〉

le 3 Im〈Q(t h Dt)u E2(t)u〉 + C(h2Dt u

2+ψ2(h)E(t)u

for u isin Cinfin

0 (RCN ) Since E ge c K ge 0 and h ψ(h) 1 when h rarr 0 we obtain (619) with scalarB = E2 for 1 and h 1

To finish the proof of Proposition 62 we substitute φwj u in the estimate (619) with Q = Q j to obtain

ψ(h)φwj u2+ 〈K jφ

wj u φwj u〉 le Im〈φwj Q j (t h Dt)u B j (t)φwj u〉 + Ch2

φwj Dt u2 (622)

for u isin Cinfin

0 (Rn+1CN ) since φwj (x Dx) and Q j (t h Dt) commute Next we shall replace the approxi-

mation Q j by the original operator Q In a G neighborhood of suppφ j we may use the Taylor expansionin w to write for almost all t

Q(t τ h12w)minus Q j (t τ )= i(K (t h12w)minus K j (t))+ (M(t h12w)minus M j (t))τ (623)

We shall start by estimating the last term in (623) Since M(t w) isin Cinfin

b we have

|M(t h12w)minus M j (t)| le Ch12 Hminus12 in suppφ j (624)

because then |wminusw j | le cHminus12 Since M(t h12w)isin S(1 h|dw|2) and h H we find from (624) that

M(t h12w)minus M j (t) isin S(h12 Hminus12G) in suppφ j uniformly in t By the CauchyndashSchwarz inequalitywe find

|〈φwj (Mw

minus M j )h Dt u B j (t)φwj u〉| le C(χwj h Dt u2+ h Hminus1

φwj u2) (625)

for u isin Cinfin

0 (Rn+1CN ) where χwj = hminus12 H 12φwj (M

wminus M j ) isin Op S(1G) uniformly in t with values

in `2 Thus we find from (616) thatsumj

χwj h Dt u2le Ch Dt u

2 u isin Cinfin

0 (Rn+1)

and for the last terms in (625) we have

h Hminus1sum

φwj u2le 2h Hminus1

u2 ψ(h)u

2 h rarr 0 u isin Cinfin

0 (Rn+1)

by (615) For the first term in the right hand side of (623) we find from Taylorrsquos formula

K (t h12w)minus K j (t)= h12〈S j (t)W j (w)〉 + R j (t τ w) in suppφ j

where S j (t) = partwK (t h12w j ) isin Linfin(R) R j isin S(h Hminus1G) uniformly for almost all t and W j isin

S(hminus12 h|dw|2) such that φ j (w)W j (w)= φ j (w)(wminusw j )= O(Hminus12) Here we could take W j (w)=

χ(h12(wminusw j ))(wminusw j ) for a suitable cut-off function χ isin Cinfin

0 We obtain from the calculus that

φwj K j (t)= φwj Kw(t h12x h12 Dx)minus h12φwj 〈S j (t)Wwj 〉 + Rwj

366 NILS DENCKER

where R j j isin S(h Hminus1G) with values in `2 for almost all t Thus we may estimate the sum of theseerror terms by (616) to obtainsum

|〈Rwj u B j (t)φwj u〉| le Ch Hminus1u

2 ψ(h)u

2 as h rarr 0 for u isin Cinfin

0 (Rn+1CN ) (626)

Observe that it follows from (617) for any κ gt 0 and almost all t that

|〈S j (t)u u〉| le C〈K j (t)u u〉12

u le κ〈K j (t)u u〉 + Cu2κ forall u isin CN

Let F j (t)= F(t h12w j )= K j (t)+ψ(h) IdN then by taking κ = H 12hminus12 we find that for any gt 0there exists h gt 0 so that

h12 Hminus12|〈S j u u〉| le 〈K j u u〉 + Ch Hminus1

u2 le 〈F j u u〉 forall u isin CN 0lt h le h (627)

since h Hminus1 ψ(h) when h 1 Now F j and S j only depend on t so by (627) we may use Remark

A6 in the Appendix for fixed t with A = h12 Hminus12S j B = F j u replaced with φwj u and v withB j H 12φwj Ww

j u Integration then gives

h12|〈B jφ

wj 〈S j (t)Ww

j 〉u φwj u〉| le32(〈F j (t)φwj u φwj u〉 + 〈F j (t)ψwj u ψwj u〉) (628)

for u isin Cinfin

0 (Rn+1CN ) 0lt h le h where

ψwj = B j H 12φwj Wwj isin Op S(1G) with values in `2

In fact since φ j isin S(1G) and W j isin S(hminus12 h|dw|2) we find that

φwj Wwj = (φ j W j )

w modulo Op S(H 12G)

Also since |φ j W j | le C Hminus12 we find from Leibnizrsquo rule that φ j W j isin S(Hminus12G) Now F ge

ψ(h) IdN h Hminus1 IdN so by using Proposition A9 in the Appendix and then integrating in t we findthat sum

〈F j (t)ψwj u ψwj u〉 le Csum

〈F j (t)φwj u φwj u〉 u isin Cinfin

0 (Rn+1CN )

We obtain from (615) that

ψ(h)u2le 2

0 (Rn+1CN )

Thus for any gt 0 we obtain from (622) and (625)ndash(628) that

(1 minus C0)sum

〈F j (t)φwj u φwj u〉 le

Im〈φwj Qu B j (t)φwj u〉 + Ch2Dt u

2 0lt h le h

We have thatsum

j B jφwj φ

wj isin S(1G) is a scalar symmetric operator uniformly in t When = 12C0

we obtain the estimate (611) with Bw = 4sum

j B jφwj φ

wj which finishes the proof of Proposition 62

Appendix

We shall first study the condition for the one-dimensional model operator

h Dt IdN +i F(t) 0 le F(t) isin Cinfin(R)

to be of finite type of order micro|δ(F)| le Cδmicro 0lt δ 1 (A1)

and we shall assume that 0 isin6infin(P) When F(t) isin Cinfin(R) we may have any microgt 0 in (A1) for examplewith F(t) = |t |1micro IdN But when F isin C1

b the estimate cannot hold with micro gt 1 and since it triviallyholds for micro= 0 the only interesting cases are 0lt microle 1

When 0 le F(t) is diagonalizable for any t with eigenvalues λ j (t) isin Cinfin j = 1 N then condi-tion (A1) is equivalent to ∣∣δ(λ j )

∣∣ le Cδmicro forall j 0lt δ 1

since δ(F)=⋃

j δ(λ j ) Thus we shall start by studying the scalar case

Proposition A1 Assume that 0le f (t)isinCinfin(R) such that f (t)ge cgt0 when |t |1 that is 0 isin6infin( f )We find that f satisfies (A1) with micro gt 0 if and only if microle 1k for an even k ge 0 so thatsum

|partj

t f (t)|gt 0 forall t (A2)

Simple examples as f (t)= eminust2show that the condition that 0 isin6infin( f ) is necessary for the conclusion

of Proposition A1

Proof Assume that (A2) does not hold with k le 1micro then there exists t0 such that f ( j)(t0) = 0 forall integer j le 1micro Then Taylorrsquos formula gives that f (t) le c|t minus t0|k and |δ( f )| ge cδ1k wherek = [1micro] + 1gt 1micro which contradicts condition (A1)

Assume now that condition (A2) holds for some k then f minus1(0) consists of finitely many points Infact else there would exist t0 where f vanishes of infinite order since f (t) 6= 0 when |t | 1 Alsonote that

⋂δgt0δ( f ) = f minus1(0) in fact f must have a positive infimum outside any neighborhood of

f minus1(0) Thus in order to estimate |δ( f )| for δ 1 we only have to consider a small neighborhood ωof t0 isin f minus1(0) Assume that

f (t0)= f prime(t0)= middot middot middot = f ( jminus1)(t0)= 0 and f ( j)(t0) 6= 0

for some j le k Since f ge 0 we find that j must be even and f ( j)(t0) gt 0 Taylorrsquos formula gives asbefore f (t)ge c|t minus t0| j for |t minus t0| 1 and thus we find that

|δ( f )capω| le Cδ1jle Cδ1k 0lt δ 1

if ω is a small neighborhood of t0 Since f minus1(0) consists of finitely many points we find that (A1) issatisfied with micro= 1k for an even k

So if 0 le F isin Cinfin(R) is Cinfin diagonalizable system and 0 isin6infin(P) condition (A1) is equivalent tosumjlek

|partj

t 〈F(t)u(t) u(t)〉|u(t)2 gt 0 forall t

368 NILS DENCKER

for any 0 6= u(t) isin Cinfin(R) since this holds for diagonal matrices and is invariant This is true also in thegeneral case by the following proposition

Proposition A2 Assume that 0 le F(t) isin Cinfin(R) is an N times N system such that 0 isin 6infin(F) We findthat F satisfies (A1) with micro gt 0 if and only if microle 1k for an even k ge 0 so thatsum

|partj

t 〈F(t)u(t) u(t)〉|u(t)2 gt 0 forall t (A3)

Observe that since 0 isin6infin(F) it suffices to check condition (A3) on a compact interval

Proof First we assume that (A1) holds with micro gt 0 let u(t) isin Cinfin(RCN ) such that |u(t)| equiv 1 andf (t)= 〈F(t)u(t) u(t)〉 isin Cinfin(R) Then we have δ( f )subδ(F) so (A1) gives

|δ( f )| le |δ(F)| le Cδmicro 0lt δ 1

The first part of the proof of Proposition A1 then gives (A3) for some k le 1microFor the proof of the sufficiency of (A3) we need the following simple lemma

Lemma A3 Assume that F(t) = Flowast(t) isin Ck(R) is an N times N system with eigenvalues λ j (t) isin Rj = 1 N Then for any t0 isin R there exist analytic v j (t) isin CN j = 1 N so that v j (t0) is abase for CN and ∣∣λ j (t)minus 〈F(t)v j (t) v j (t)〉

∣∣ le C |t minus t0|k for |t minus t0| le 1

after a renumbering of the eigenvalues

By a well-known theorem of Rellich the eigenvalues λ(t) isin C1(R) for symmetric F(t) isin C1(R)see [Kato 1966 Theorem II68]

Proof It is no restriction to assume t0 = 0 By Taylorrsquos formula

F(t)= Fk(t)+ Rk(t)

where Fk and Rk are symmetric Fk(t) is a polynomial of degree k minus 1 and Rk(t)= O(|t |k) Since Fk(t)is symmetric and holomorphic it has a base of normalized holomorphic eigenvectors v j (t) with realholomorphic eigenvalues λ j (t) by [Kato 1966 Theorem II61] Thus λ j (t)= 〈Fk(t)v j (t) v j (t)〉 and bythe minimax principle we may renumber the eigenvalues so that

|λ j (t)minus λ j (t)| le Rk(t) le C |t |k forall j

The result then follows since

|〈(F(t)minus Fk(t))v j (t) v j (t)〉| = |〈Rk(t)v j (t) v j (t)〉| le C |t |k forall j

Assume now that Equation (A3) holds for some k As in the scalar case we have that k is even and⋂δgt0δ(F)=60(F)= |F |

minus1(0) Thus for small δ we only have to consider a small neighborhood oft0 isin60(F) Then by using Lemma A3 we have after renumbering that for each eigenvalue λ(t) of F(t)there exists v(t) isin Cinfin so that |v(t)| ge c gt 0 and

Now if 60(F) 3 t j rarr t0 is an accumulation point then after choosing a subsequence we obtain thatfor some eigenvalue λk we have λk(t j ) = 0 for all j Then λk vanishes of infinite order at t0 contra-dicting (A3) by (A4) Thus we find that 60(F) is a finite collection of points By using (A3) withu(t)= v(t) we find as in the second part of the proof of Proposition A1 that

〈F(t)v(t) v(t)〉 ge c|t minus t0| j|t minus t0| 1

for some even j le k which by (A4) gives that

λ(t)ge c|t minus t0| jminus C |t minus t0|k+1

ge cprime|t minus t0| j

|t minus t0| 1

Thus |δ(λ) cap ω| le cδ1j if ω for δ 1 if ω is a small neighborhood of t0 isin 60(F) Since δ(F) =⋃j δ(λ j ) where λ j (t) j are the eigenvalues of F(t) we find by adding up that |δ(F)| le Cδ1k

Thus the largest micro satisfying (A1) must be ge 1k

Let A(t) isin Lip(RL(L2(Rn))) be the L2(Rn) bounded operators which are Lipschitz continuous inthe parameter t isin R This means that

A(s)minus A(t)s minus t

= B(s t) isin L(L2(Rn)) uniformly in s and t (A5)

One example is A(t) = aw(t x Dx) where a(t x ξ) isin Lip(R S(1G)) for a σ temperate metric Gwhich is constant in t such that GGσ

Lemma A4 Assume that A(t)isin Lip(RL(L2(Rn))) and φ(τ)isinCinfin(R) such that φprime(τ )isinCinfin

0 (R) Thenfor κ gt 0 we can estimate the commutator∥∥[

φ(Dtκ) A(t)]∥∥

L(L2(Rn+1))le Cκminus1

where the constant only depends on φ and the bound on A(t) in Lip(RL(L2(Rn)))

Proof In the following we shall denote by A(t x y) the distribution kernel of A(t) Then we findfrom (A5) that

A(s x y)minus A(t x y)= (s minus t)B(s t x y) (A6)

where B(s t x y) is the kernel for B(s t) for s t isin R Thenlang[φ(Dtκ) A(t)

rang= (2π)minus1

intei(tminuss)τφ(τκ)(A(s x y)minus A(t x y))u(s x)v(t y) dτ ds dt dx dy (A7)

for u v isin Cinfin

0 (Rn+1) and by using (A6) we obtain that the commutator has kernel

intei(tminuss)τφ(τκ)(sminust)B(stxy)dτ =

intei(tminuss)τρ(τκ)B(stxy)dτ = ρ(κ(sminust))B(stxy)

in D(R2n+2) where ρ isin Cinfin

0 (R) Thus we may estimate (A7) by using CauchyndashSchwarzint|(κs)〈B(s + t t)u(s + t) v(t)〉L2(Rn)| dt ds le Cκminus1

where the norms are in L(L2(Rn+1))

370 NILS DENCKER

We shall need some results about the lower bounds of systems and we shall use the following versionof the Garding inequality for systems A convenient way for proving the inequality is to use the Wickquantization of a isin Linfin(T lowastRn) given by

aWick(x Dx)u(x)=

intT lowastRn

a(y η)6wyη(x Dx)u(x) dy dη u isin S(Rn)

using the rank one orthogonal projections 6wyη(x Dx) in L2(Rn) with Weyl symbol

6yη(x ξ)= πminusn exp(minus|x minus y|

2minus |ξ minus η|2

)(see [Dencker 1999 Appendix B] or [Lerner 1997 Section 4]) We find that aWick S(Rn) 7rarr Sprime(Rn) issymmetric on S(Rn) if a is real-valued

a ge 0 HrArr(aWick(x Dx)u u

)ge 0 u isin S(Rn) (A8)

aWick(x Dx)L(L2(Rn)) le aLinfin(T lowastRn)

which is the main advantage with the Wick quantization If a isin S(1 h|dw|2) we find that

aWick= aw + rw (A9)

where r isin S(h h|dw|2) For a reference see [Lerner 1997 Proposition 42]

Proposition A5 Let 0 le A isin Cinfin

b (TlowastRn) be an N times N system then we find that

〈Aw(x h D)u u〉 ge minusChu2

forall u isin Cinfin

0 (RnCN )

This result is well known (see for example Theorem 18614 in Volume III of [Hormander 1983ndash1985])but we shall give a short and direct proof

Proof By making a L2 preserving linear symplectic change of coordinates (x ξ) 7rarr (h12x hminus12ξ)

we may assume that 0 le A isin S(1 h|dw|2) Then we find from (A9) that Aw = AWick

+ Rw whereR isin S(h h|dw|

2) Since A ge 0 we obtain from (A8) that

〈Awu u〉 ge 〈Rwu u〉 ge minusChu2

0 (RnCN )

Remark A6 Assume that A and B are N times N matrices such that plusmnA le B Then we find

|〈Au v〉| le32 (〈Bu u〉 + 〈Bv v〉) forall u v isin Cn

In fact since B plusmn A ge 0 we find by the CauchyndashSchwarz inequality that

2 |〈(B plusmn A)u v〉| le 〈(B plusmn A)u u〉 + 〈(B plusmn A)v v〉 forall u v isin Cn

and 2 |〈Bu v〉| le 〈Bu u〉 + 〈Bv v〉 The estimate can then be expanded to give the inequality since

|〈Au u〉| le 〈Bu u〉 forall u isin Cn

by the assumption

Lemma A7 Assume that 0 le F(t) isin C2(R) is an N times N system such that F primeprimeisin Linfin(R) Then we have

|〈F prime(0)u u〉|2le CF primeprime

Linfin〈F(0)u u〉u2

forall u isin CN

Proof Take u isin CN with |u| = 1 and let 0 le f (t) = 〈F(t)u u〉 isin C2(R) Then | f primeprime| le F primeprime

Linfin soLemma 772 in Volume I of [Hormander 1983ndash1985] gives

| f prime(0)|2 = |〈F prime(0)u u〉|2le CF primeprime

Linfin f (0)= CF primeprimeLinfin〈F(0)u u〉

Lemma A8 Assume that F ge 0 is an N times N matrix and that A is a L2 bounded scalar operator Then

0 le 〈F Au Au〉 le A2〈Fu u〉

0 (RnCN )

Proof Since F ge 0 we can choose an orthonormal base for CN such that 〈Fu u〉 =sumN

j=1 f j |u j |2 for

u = (u1 u2 ) isin CN where f j ge 0 are the eigenvalues of F In this base we find

0 le 〈F Au Au〉 =

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

Proposition A9 Assume that hH le F isin S(1 g) is an N times N system φ j and ψ j isin S(1G) withvalues in `2 such that

sumj |φ j |

2ge c gt 0 and ψ j is supported in a fixed G neighborhood of w j isin suppφ j

for all j Here g = h|dw|2 and G = H |dw|

2 are constant metrics 0 lt h le H le 1 If F j = F(w j ) wefind for H 1 thatsum

〈F jψwj (x Dx)u ψwj (x Dx)u〉 le C

〈F jφwj (x Dx)u φwj (x Dx)u〉 (A10)

0 (RnCN )

Proof Since χ =sum

j |φ j |2ge c gt 0 we find that χminus1

isin S(1G) The calculus gives

(χminus1)wsum

φwj φwj = 1 + rw

where r isin S(HG) uniformly in H Thus the mapping u 7rarr (χminus1)wsum

j φwj φ

wj u is a homeomorphism

on L2(Rn) for small enough H Now the constant metric G = H |dw|2 is trivially strongly σ temperate

according to Definition 71 in [Bony and Chemin 1994] so Theorem 76 in the same reference givesB isin S(1G) such that

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

where Bwj = Bw(χminus1)wφwj isin Op S(1G) uniformly which gives 1 =sum

j φwj B

j since (Bwj )lowast= B

j Nowwe shall put

Fw(x Dx)=

ψwj (x Dx)F jψwj (x Dx)

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

wl Bwk then we find from (A11) that

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

Let d jk be the Hminus1|dw|

2 distance between the G neighborhoods in which ψ j and ψk are supportedThe usual calculus estimates (see [Hormander 1983ndash1985 Volume III page 168] or [Bony and Chemin1994 Theorem 26]) gives that the L2 operator norm of Cw

jkl can be estimated by

Cwjkl le CN (1 + d jl + dlk)

minusN

for any N We find by Taylorrsquos formula Lemma A7 and the CauchyndashSchwarz inequality that

|〈(F j minus Fk)u u〉| le C1|w j minuswk |〈Fku u〉12h12

u + C2h|w j minuswk |2u

2le C〈Fku u〉(1 + d jk)

since |w j minuswk | le C(d jk + Hminus12) and h le h Hminus1le Fk Since Fl ge 0 we obtain that

2|〈Flu v〉| le 〈Flu u〉12

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

jklφwk u〉 le C jkl

2〈Fkφ

wk u φwk u〉

Thus we find thatsumjkl

〈FlCwjklφ

wk u φwj u〉 le CN

sumjkl

(1 + d jl + dlk)2minusN

〈Fkφwk u φwk u〉

12〈F jφ

wj u φwj u〉

sumjkl

(1 + d jl)1minusN2(1 + dlk)

1minusN2(〈F jφ

wj u φwj u〉 + 〈Fkφ

wk u φwk u〉

)Since

sumj (1 + d jk)

minusNle C for all k for N large enough by [Hormander 1983ndash1985 Volume III page

168]) we obtain the estimate (A10) and the result

References

[Bony and Chemin 1994] J-M Bony and J-Y Chemin ldquoEspaces fonctionnels associeacutes au calcul de Weyl-Houmlrmanderrdquo BullSoc Math France 1221 (1994) 77ndash118 MR 95a35152 Zbl 079835172

[Davies 2002] E B Davies ldquoNon-self-adjoint differential operatorsrdquo Bull London Math Soc 345 (2002) 513ndash532 MR2003c47081 Zbl 105247042

[Dencker 1982] N Dencker ldquoOn the propagation of polarization sets for systems of real principal typerdquo J Funct Anal 463(1982) 351ndash372 MR 84c58081 Zbl 048758028

[Dencker 1993] N Dencker ldquoPreparation theorems for matrix valued functionsrdquo Ann Inst Fourier (Grenoble) 433 (1993)865ndash892 MR 95f32009 Zbl 078358010

[Dencker 1999] N Dencker ldquoA sufficient condition for solvabilityrdquo Internat Math Res Notices 12 (1999) 627ndash659 MR2000e35245 Zbl 094758019

[Dencker et al 2004] N Dencker J Sjoumlstrand and M Zworski ldquoPseudospectra of semiclassical (pseudo-) differential opera-torsrdquo Comm Pure Appl Math 573 (2004) 384ndash415 MR 2004k35432 Zbl 105435035

[Dimassi and Sjoumlstrand 1999] M Dimassi and J Sjoumlstrand Spectral asymptotics in the semi-classical limit London Math SocLecture Note Series 268 Cambridge University Press 1999 MR 2001b35237 Zbl 092635002

[Elsner 1985] L Elsner ldquoAn optimal bound for the spectral variation of two matricesrdquo Linear Algebra Appl 71 (1985) 77ndash80MR 87c15035 Zbl 058315009

[Helffer and Sjoumlstrand 1990] B Helffer and J Sjoumlstrand ldquoAnalyse semi-classique pour lrsquoeacutequation de Harper II Comportementsemi-classique pregraves drsquoun rationnelrdquo Meacutem Soc Math France (NS) 40 (1990) 139pp MR 91h35018 Zbl 071434131

[Houmlrmander 1983ndash1985] L Houmlrmander The analysis of linear partial differential operators IndashIV Grundlehren der MathWissenschaften 256ndash257 274ndash275 Springer-Verlag Berlin 1983ndash1985

[Kato 1966] T Kato Perturbation theory for linear operators Die Grundlehren der math Wissenschaften 132 Springer NewYork 1966 MR 34 3324 Zbl 014812601

[Lerner 1997] N Lerner ldquoEnergy methods via coherent states and advanced pseudo-differential calculusrdquo pp 177ndash201 inMultidimensional complex analysis and partial differential equations (Satildeo Carlos 1995) edited by P D Cordaro et alContemp Math 205 Amer Math Soc 1997 MR 98d35244 Zbl 088535152

[Martinez 2002] A Martinez An introduction to semiclassical and microlocal analysis Springer New York 2002 MR 2003b35010 Zbl 099435003

[Pravda-Starov 2006a] K Pravda-Starov ldquoA complete study of the pseudo-spectrum for the rotated harmonic oscillatorrdquo JLondon Math Soc (2) 733 (2006) 745ndash761 MR 2007c34133 Zbl 110634060

[Pravda-Starov 2006b] K Pravda-Starov Etude du pseudo-spectre drsquoopeacuterateurs non auto-adjoints PhD thesis Universiteacute deRennes I 2006

[Sjoumlstrand 1982] J Sjoumlstrand Singulariteacutes analytiques microlocales Asteacuterisque 95 Socieacuteteacute matheacutematique de France Paris1982 MR 84m58151 Zbl 052435007

[Taylor 1981] M E Taylor Pseudodifferential operators Princeton Math Series 34 Princeton University Press 1981 MR82i35172 Zbl 045347026

[Trefethen and Embree 2005] L N Trefethen and M Embree Spectra and pseudospectra The behavior of nonnormal matricesand operators Princeton University Press 2005 MR 2006d15001 Zbl 108515009

Received 19 Feb 2008 Revised 25 Sep 2008 Accepted 22 Oct 2008

NILS DENCKER Lund University Centre for Mathematical Sciences Box 118 SE-221 00 Lund Swedendenckermathslthsehttpwwwmathslthsematematiklupersonaldenckerhomepagehtml

1 Introduction

2 The definitions

3 The interior case

Appendix

References

ANALYSIS AND PDEVol 1 No 3 2008

NILS DENCKER

The pseudospectrum (or spectral instability) of non-self-adjoint operators is a topic of current interest inapplied mathematics In fact for non-self-adjoint operators the resolvent could be very large outside thespectrum making numerical computation of the complex eigenvalues very hard This has importancefor example in quantum mechanics random matrix theory and fluid dynamics

The occurrence of false eigenvalues (or pseudospectrum) of non-self-adjoint semiclassical differentialoperators is due to the existence of quasimodes that is approximate local solutions to the eigenvalueproblem For scalar operators the quasimodes appear generically since the bracket condition on theprincipal symbol is not satisfied for topological reasons

In this paper we shall investigate how these results can be generalized to square systems of semiclas-sical differential operators of principal type These are the systems whose principal symbol vanishes offirst order on its kernel We show that the resolvent blows up as in the scalar case except in a nowheredense set of degenerate values We also define quasisymmetrizable systems and systems of subelliptictype for which we prove estimates on the resolvent

1 Introduction

In this paper we shall study the pseudospectrum or spectral instability of square non-self-adjoint semi-classical systems of principal type Spectral instability of non-self-adjoint operators is currently a topicof interest in applied mathematics see [Davies 2002] and [Trefethen and Embree 2005] It arises fromthe fact that for non-self-adjoint operators the resolvent could be very large in an open set containing thespectrum For semiclassical differential operators this is due to the bracket condition and is connectedto the problem of solvability In applications where one needs to compute the spectrum the spectralinstability has the consequence that discretization and round-off errors give false spectral values so-called pseudospectra see [Trefethen and Embree 2005] and references there

We shall consider bounded systems P(h) of semiclassical operators given by (22) and we shallgeneralize the results of the scalar case in [Dencker et al 2004] Actually the study of unboundedoperators can in many cases be reduced to the bounded case see Proposition 220 and Remark 221We shall also study semiclassical operators with analytic symbols in the case when the symbols can beextended analytically to a tubular neighborhood of the phase space satisfying (23) The operators westudy will be of principal type which means that the principal symbol vanishes of first order on thekernel see Definition 31

The definition of semiclassical pseudospectrum in [Dencker et al 2004] is essentially the bracketcondition which is suitable for symbols of principal type By instead using the definition of (injectivity)

MSC2000 primary 35S05 secondary 58J40 47G30 35P05Keywords pseudospectrum spectral instability system semiclassical non-self-adjoint pseudodifferential operator

324 NILS DENCKER

where 1 = minussumn

j=1 part2x j

2 The definitions

Pw(x h Dx)u =1

(2π)n

intintT lowastRn

P( x + y

P(h)sim

infinsumj=0

with Pj isin Cinfin

326 NILS DENCKER

Example 21 Let

P(ξ)=

) ξ isin R

= minus1z(

1 ξz0 1

le C |w| 1 (24)

k ge 1

⋃jgt1

part j (P)

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

Example 28 Let

(0 1t 0

)t isin R

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

6ss(P)=λ πminus1

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

324 NILS DENCKER

where 1 = minussumn

j=1 part2x j

2 The definitions

Pw(x h Dx)u =1

(2π)n

intintT lowastRn

P( x + y

P(h)sim

infinsumj=0

with Pj isin Cinfin

326 NILS DENCKER

Example 21 Let

P(ξ)=

) ξ isin R

= minus1z(

1 ξz0 1

le C |w| 1 (24)

k ge 1

⋃jgt1

part j (P)

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

Example 28 Let

(0 1t 0

)t isin R

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

6ss(P)=λ πminus1

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

2 The definitions

Pw(x h Dx)u =1

(2π)n

intintT lowastRn

P( x + y

P(h)sim

infinsumj=0

with Pj isin Cinfin

326 NILS DENCKER

Example 21 Let

P(ξ)=

) ξ isin R

= minus1z(

1 ξz0 1

le C |w| 1 (24)

k ge 1

⋃jgt1

part j (P)

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

Example 28 Let

(0 1t 0

)t isin R

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

6ss(P)=λ πminus1

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

326 NILS DENCKER

Example 21 Let

P(ξ)=

) ξ isin R

= minus1z(

1 ξz0 1

le C |w| 1 (24)

k ge 1

⋃jgt1

part j (P)

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

Example 28 Let

(0 1t 0

)t isin R

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

6ss(P)=λ πminus1

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

k ge 1

⋃jgt1

part j (P)

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

Example 28 Let

(0 1t 0

)t isin R

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

6ss(P)=λ πminus1

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

328 NILS DENCKER

Example 27 If

(λ1(w) 1

0 λ2(w)

1(P)= (w λ) λ= λ j (w) j = 1 2

2(P)= (w λ) λ= λ1(w)= λ2(w)

Example 28 Let

(0 1t 0

)t isin R

Example 29 If

(w1 +w2 w3

w3 w1 minusw2

1(P)=(w λ j ) λ j = w1 + (minus1) j

radicw2

2 +w23 j = 1 2

6ss(P)=λ πminus1

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

6ws(P)=6(P)= R

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

330 NILS DENCKER

Cinfin

minus1isin Cinfin

isin Cinfin

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

αmminus1(w) |w| 1

le C forallw

ζ minus 1IdN

)minus1ζ 6= 1

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

332 NILS DENCKER

minus1isin Cinfin

forall k (28)

WFh(u j )

A(WFpolh (u))=

sube WFpolh (A(h)u)

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

for any v isin CN

3scmicro (P(h))=

λscmicro (P(h)) =

⋂micro3

3 The interior case

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

334 NILS DENCKER

AP B =

(λ 00 E

(λ1(w) 1

0 λ2(w)

(w1 minusw2 w2

w2 minusw1 minusw2

partw1 P =

(1 00 minus1

(w0 λ0) isin4(P)

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

w21 +w2

1 so dw p = 0 when w1 = λ= 0

partkν |P(w0)| 6= 0 k = κP(w0 0) (33)

AP(w)B =

(P11(w) P12(w)

P21(w) P22(w)

me(w0 0)

Example 39 Let

P(x ξ)=

(q(x ξ) χ(x)

0 q(x ξ)

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

336 NILS DENCKER

)sube3minus(P) (35)

ge C prime

3minus(P)=3(P)=

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

) W = (3minus(P) W )cup (3+(P) W )

) (38)

Elowast(w)P(w)E(w)=

(λ(w) IdK P12

)= P0(w) (39)

(Elowast)wP(h)Ew =

(P11 P12

P21 P22

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

338 NILS DENCKER

0 IdNminusK

sumjge1

h j Pwj (x h D) (310)

A(h) =sum

sumjge1

h j Ewj (x h D)

Ek =12i

B1 minus A1 =

HλA0 + P1 A0

)λminus1

isin Cinfin

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

Example 314 Let

(λw(x h D) 1

h λw(x h D)

(λ(w) 1

0 λ(w)

340 NILS DENCKER

sube3minus(P) (311)

le Ch h rarr 0 (41)

2(0) gt 0 and let

Pw(t h Dt)=

)= Pw(t h Dt)

lowast

+ a21(t)+ a2

2(t) isin R

(1 i1 minusi

(1 1i minusi

(h Dt + ia2(t) 0

)= P(h)

radic2nh

Pw(t h Dt)=

(h Dt a(t)

minusha(t) h Dt

P(t τ )=

(τ a(t)0 τ

h12 00 minus1

(hminus12 0

radich

(radich Dt a(t)

a(t) minusradic

ge exp(cradic

342 NILS DENCKER

(1 h Dt

h iha(t)

1 0minush 1

(1 minush Dt

(1 00 minush

2 c gt 0 (43)

12(A +

perp (47)

344 NILS DENCKER

Ran MPperp Ker MP

P(w0)=

(0 P12(w0)

0 P22(w0)

(M11 M12

M21 M22

we find

(0 00 M22 P22

)at w0

22 P22 at w0

2 c gt 0 (48)

Im〈Qu u〉 ge 0 (49)

for any u isin CN

2ge minusεuprime

2minus C prime

forall ε gt 0

ge minusC primeQu

u2 u isin Ker QE

Ker Qlowast

E = Ker QE

346 NILS DENCKER

P(w)= Bminus1(w)

(λ(w) IdK 0

0 P22(w)

)B(w) (411)

|d Re q(λminus z)| 6= 0 (412)

M = Blowast

(q IdK 0

(q(λminus z) IdK 0

0 IdNminusK

)B (414)

|ξ |2 = z

Example 414 Let

(w2 + iw3 w1

w1 w2 minus iw3

)w = (w1 w2 w3)

(0 11 0

M P(w)=

(w1 w2 minus iw3

w2 + iw3 w1

)= (M P(w))lowast

w21 minusw2

b (TlowastRn)

| le Ch 0lt h 1 (416)

P(x ξ)= |ξ |2 IdN +i K (x)

348 NILS DENCKER

εh adT Q(h)sim

infinsumk=0

hkk(adT )

k(Q(h))

u le Qεu (423)

u leCε

Pεu (425)

0 K h log 1h

)cap Spec(P)= empty

P(h)sim

sumjge0

350 NILS DENCKER

given by

2)(2h)u(y) dy (426)

u2H(3T )

2L2(h)

Pw(x h D)u2H(3T )

= (P|3T )T3T u2L2(h) + O(h)u

2H(3T )

of P gives

P(w+ iHT )= P(w)+ iHT P(w)+ O(2) || 1

for δ small enough

0 (51)

|partj

t f | 6= 0 (52)

(t2 t3

Then we have

F (3)(0)=

(0 66 0

352 NILS DENCKER

Ker F ( j)(0)= 0

ch Observe that

(1 minustt 1

) (t2 00 0

) (1 tminust 1

+ t8 t3minus t7

t3minus t7 t4

(1 minustt 1

) (t2 00 t6

) (1 tminust 1

Then we have that

P = (1 + t2)minus1(

1 minustt 1

) (h Dt + i(t2

+ t4) 00 h Dt + i(t6

) (1 tminust 1

)+ O(h)

δ(F)=

0lt δ le 1

where C = Eminus1

Example 59 Let

P = h Dt Id2 +αh(

Dx 00 minusDx

5(w)=1

int|z|=ε

354 NILS DENCKER

(Q11 Q12

220 IdNminusK

∣∣ le Cφ(δ) 0lt δ 1 (57)

in Definition 515

(Q11 Q12

(Q11 00 Q22

356 NILS DENCKER

(Q11 00 Q22

j Q j j B j with

(Blowast

1 B1 00 Blowast

)we obtain that

Q = BQ = BM P =

(Blowast

1 Q11 B1 00 Blowast

2 Q22 B2

(Q11 00 Q22

(minusQlowast

11 00 minusQlowast

if and only if

|partkt f |gt 0

on the resolvent

358 NILS DENCKER

|partj

∣∣ le Cφ(δ) 0lt δ 1 (64)

0 (66)

0 (67)

sumjge1 φ j

0 (69)

(Q11 00 Q22

at w0 we find

uprimeprime le C(Qw

isin Cinfin

0 (RnCNminusK )

+Qw22uprimeprime

360 NILS DENCKER

u near (0 w0) (610)

Qw= Ew0 Qw

0 + Rw0 + h Rw1

sim= Mw0 ((Id +i Mw

1 )h Dt + i Kw1 )M

w0 modulo O(h)

g(ν j )12

Dt u2 0lt h 1 (611)

radicψ Next we

χ prime

)(613)

j isin Cinfin

∥∥L(L2(Rn+1))

le Cradicψε

362 NILS DENCKER

2 h 1 (614)

radicψ

radicψ and

radicψQuu + hu

sumj φ

2j equiv 1

2) Sincesumφ2

u2le u

for u isin Cinfin

φwj (x Dx)u2le 2u

0 (Rn) (615)

rwj u2le C H 2ν

u2 (616)

〈K j (t)u u〉 le δ∣∣∣ le Cφ(δ) (618)

Dt u2 0lt h 1 (619)

364 NILS DENCKER

Letting

E(t)= exp(ψ(h)

08h(s) ds

2 u isin Cinfin

0 (RCN )

Dt u2+ hu

0 (RCN ) (621)

we obtain

2+ C0hu

0 (RCN )

2+ψ2(h)E(t)u

for u isin Cinfin

φwj Dt u2 (622)

for u isin Cinfin

b we have

|〈φwj (Mw

φwj u2) (625)

for u isin Cinfin

2 u isin Cinfin

0 (Rn+1)

h Hminus1sum

u2 ψ(h)u

0 (Rn+1)

366 NILS DENCKER

2 ψ(h)u

0 (Rn+1CN ) (626)

|〈S j (t)u u〉| le C〈K j (t)u u〉12

h12|〈B jφ

wj 〈S j (t)Ww

for u isin Cinfin

0 (Rn+1CN )

ψ(h)u2le 2

0 (Rn+1CN )

(1 minus C0)sum

2 0lt h le h

We have thatsum

j B jφwj φ

Appendix

since δ(F)=⋃

|partj

of Proposition A1

|partj

368 NILS DENCKER

|partj

F(t)= Fk(t)+ Rk(t)

|t minus t0| 1

rang= (2π)minus1

for u v isin Cinfin

370 NILS DENCKER

aWick(x Dx)u(x)=

intT lowastRn

aWick= aw + rw (A9)

0 (RnCN )

by the assumption

forall u isin CN

0 (RnCN )

j=1 f j |u j |2 for

f jAu j2le A

f ju j2= A

2〈Fu u〉

for u isin Cinfin

0 (RnCN )

sumj |φ j |

0 (RnCN )

Proof Since χ =sum

(χminus1)wsum

φwj φwj = 1 + rw

j φwj φ

Bw(χminus1)wsum

φwj φwj =

Bwj φwj = 1

j φwj B

j Nowwe shall put

Fw(x Dx)=

ThenFw

φwj Bw

j FwBwk φwk =

sumjkl

φwj Bw

j ψwl Flψ

wl Bwk φ

wk (A11)

372 NILS DENCKER

Let Cwjkl = B

j ψwl ψ

〈Fwu u〉 =

sumjkl

〈FlCwjklφ

wk u φwj u〉

minusN

〈Flv v〉12

le C〈F j u u〉12

〈Fkv v〉12(1 + d jl)(1 + dlk)

and Lemma A8 gives

〈FkCwjklφ

wk u FkCw

2〈Fkφ

wk u φwk u〉

〈FlCwjklφ

sumjkl

12〈F jφ

wj u φwj u〉

sumjkl

1minusN2(〈F jφ

wk u φwk u〉

)Since

sumj (1 + d jk)

References

1 Introduction

2 The definitions

3 The interior case

Appendix

References

Documents

The pseudospectrum of systems of semiclassical operators