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BULLETIN (New Series) OF THEAMERICAN MATHEMATICAL SOCIETYVolume 49, Number 3, July 2012, Pages 415442S 0273-0979(2011)01368-4Article electronically published on December 28, 2011

UNIQUENESS PROPERTIES OF SOLUTIONS

TO SCHRODINGER EQUATIONS

L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA

1. Introduction

To place the subject of this paper in perspective, we start out with a briefdiscussion of unique continuation. Consider solutions to

(1.1) u(x) =

nj=1

2

ux2j(x) = 0

(harmonic functions) in the unit ball{x Rn :|x| 2, since harmonic functions arestill real analytic in{x Rn :|x| 2), who proved the SUCP for

P(x, D) = + V(x), with V Lloc(Rn).In order to establish his result, Carleman introduced a method (the method ofCarleman estimates) which has permeated the subject ever since. In this context,an example of a Carleman estimate is:

For f

C0 (

{x

Rn :

|x

|0 and

w(r) =r exp( r

0

es 1s

ds),

one has

(1.2) 3

w12(|x|)f2(x)dx c

w22(|x|) |f(x)|2dx,with c independent of .

Received by the editors September 16, 2011.2010 Mathematics Subject Classification. Primary 35Q55.Key words and phrases. Schrodinger evolutions.

The first and fourth authors are supported by MEC grant, MTM2004-03029, the second andthird authors by NSF grants DMS-0968472 and DMS-0800967, respectively.

c2011 American Mathematical Society415

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416 L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA

For a proof of this estimate, see [25],[7]. The SUCP of CarlemanMuller followseasily from (1.2) (see[37]for instance).

In the late 1950s and 1960s there was a great deal of activity on the subject ofSUCP and the closely related uniqueness in the Cauchy problem, some highlights

being[1] and[8], respectively, both of which use the method of Carleman estimates.These results and methods have had a multitude of applications to many areas ofanalysis, including to nonlinear problems. (For a recent example, see [38] for anapplication to energy critical nonlinear wave equations).

In connection with the CarlemanMuller SUCP a natural question is, How fastis a solution u allowed to vanish before it must vanish identically?

By considering n = 2, u(x1, x2) =(x1+ ix2)N, we see that to make sense ofthe question, a normalization is required; for instance,

sup|x|0,

then u 0.It turns out that a quantitative formulation of this can also be proved, as was

done in[7], and this was crucial for the resolution in[7] of a long-standing problemin disordered media, namely Anderson localization near the bottom of the spectrumfor the continuous AndersonBernoulli model in Rn, n 1.

Next, we turn to versions of unique continuation for evolution equations. We

start with parabolic equations and consider solutions oftu u= W u + V u, with W+ V <

(or equivalently|tu u| M(|u| + |u|)). Using a parabolic analog of theCarleman estimate described earlier, one can show that if

|tu u| M(|u| + |u|), (x, t) {x Rn : |x| 0,with|u(x)| A and

u 0, (x, t) {x Rn :R < |x|

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UNIQUE CONTINUATION 417

This result is closely related to the elliptic SUCP discussed before. On the otherhand, for parabolic equations, there is also abackward uniquenessprinciple, whichis very useful in applications to control theory (see [44] for an early result in thisdirection): Consider solutions to

|tu u| M(|u| + |u|), (x, t) Rn (0, 1],withu A. Then, if u(, 1) 0, we must have u 0. This result is alsoproved through Carleman estimates (see [44]).

Recently, a strengthening of this result has been obtained in [ 24], where oneconsiders solutions only defined in Rn+ (0, 1], Rn+= {(x1, . . . , xn) Rn : x1>0},without any assumptions on u at x1= 0, and still obtains the backward uniquenessresult. This strengthening had an important application to nonlinear equations,allowing the authors of[24]to establish a long-standing conjecture of J. Leray onregularity and uniqueness of solutions to the NavierStokes equations (see also[52]for a recent extension).

Finally, we turn to dispersive equations. Typical examples of these are the k-generalized KdV equation

(1.3) tu + 3xu + u

kxu= 0, (x, t) R R, k Z+,and the nonlinear Schrodinger equation

(1.4) tu= i(u |u|p1u), (x, t) Rn R, p >1.These equations model phenomena of wave propagation and have been extensivelystudied in the last 30 years or so.

For these equations, unique continuation through spatial boundaries also holds,as was shown by Saut-Scheurer[51] for the KdV-type equations and by Izakov[35]for Schrodinger-type equations. (All of these results were established through Car-leman estimates.) These equations however are time reversible (no preferred timedirection) and so backward uniqueness is immediate, unlike in parabolic problems.Once more in connection with control theory, this time for dispersive equations,Zhang [56]showed, for solutions of

(1.5) tu= i(2xu |u|2u), (x, t) R [0, 1],

that if u(x, t) = 0 for (x, t) (, a) {0, 1} (or (x, t) (a, ) {0, 1}) for somea R, then u 0. Zhangs proof was based on the inverse scattering method,which uses that this is a completely integrable model, and did not apply to other

nonlinearities or dimensions. This type of result was extended to the k-generalizedKdV equation in (1.3) and the general nonlinear Schrodinger equation in (1.4) in alldimensions (where inverse scattering is no longer available) using suitable Carlemanestimates (see[39], [33],[34], and references therein).

For recent surveys of the results presented so far, see [36], [37].Returning to backward uniqueness for parabolic equations, in analogy with Lan-

diss elliptic conjecture mentioned earlier, Landis and Oleinik [43]conjectured thatin the backward uniqueness result one can replace the hypothesis u(, 1) 0 withthe weaker one

|u(x, 1)| c ec|x|2+

, for some >0.

This is indeed true and was established in[18]and[49]. Similarly, one can conjecture(as was done in [20]) that for Schrodinger equations, if

|u(x, 0)| + |u(x, 1)| c ec|x|2+

, for some >0,

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then u 0. This was established in[18].In analogy with the improvement of backward uniqueness in [24], one can show

that it suffices to deal with solutions in Rn+ (0, 1] (for parabolic problems) andrequire

|u(x, 1)| c ecx2+1 , x1>0, for some >0,

to conclude that u 0 ([49]), and that for the Schrodinger equations it suffices tohave u a solution in Rn+ [0, 1], with

|u(x, 0)| + |u(x, 1)| c ecx2+1 , x1>0, for some >0,

to conclude that u 0, as we will prove in section 5 of this paper.In [16] it was pointed out for the first time (see also [10]) that both the results

in [18]and in [16]in the case of the free heat equation,

tu= u,

and the free Schrodinger equation,

tu= iu,

respectively, are in fact a corollary of the more precise Hardy uncertainty principlefor the Fourier transform, which says,

iff(x) =O(e|x|2/2),

f() =O(e4||

2/2) and1/ >1/4, then

f 0, and if1/= 1/4, f(x) =ce|x|2/2 , as will be discussedbelow.

Thus, in a series of papers ([16][23], [11]) we took up the task of finding thesharp version of the Hardy uncertainty principle, in the context of evolution equa-tions. The results obtained have already yielded new results on nonlinear equations.For instance in[21] and[23] we have found applications to the decay of concentra-tion profiles of possible self-similar type blow-up solutions of nonlinear Schrodngerequations and to the decay of possible solitary wave type solutions of nonlinearSchrodinger equations.

In the rest of this work we shall review some of our recent results concerningunique continuation properties of solutions of Schrodinger equations of the form

(1.6) tu= i(u + F(x,t,u, u)), (x, t) Rn R.We shall be mainly interested in the case where

(1.7) F(x,t,u, u) =V(x, t)u(x, t)

describes the evolution of the Schrodinger flow with a time-dependent potentialV(x, t), and in the semilinear case

(1.8) F(x,t,u,u) =F(u,u),

withF : C C C, F(0, 0) = uF(0, 0) = uF(0, 0) = 0.Let us consider a familiar dispersive model, thek-generalized Kortewegde Vries

equation (1.3), and recall a theorem established in[17]:

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Theorem 1. There existsc0>0 such that for any pair

u1, u2 C([0, 1] :H4(R) L2(|x|2dx))of solutions of (1.3), if

(1.9) u1(, 0) u2(, 0), u1(, 1) u2(, 1) L2(ec0x3/2+ dx),thenu1 u2.

Above we have used the notationx+= max{x; 0}.Notice that by taking u2 0 Theorem 1 gives a restriction on the possible

decay of a nontrivial solution of (1.3) at two different times. The power 3/2 inthe exponent in (1.9) reflects the asymptotic behavior of the Airy function. Moreprecisely, the solution of the initial value problem (IVP)

(1.10) tv+

3xv= 0,

v(x, 0) =v0(x),is given by the group{U(t) : t R}

U(t)v0(x) = 13

3tAi

3

3t

v0(x),

where

Ai(x) =c

eix+i3

d,

is the Airy function which satisfies the estimate

|Ai(x)| c(1 + x)1/4 ecx3/2+ .

It was also shown in [17]that Theorem 1 is optimal:

Theorem 2. There exists u0 S(R), u0 0 and T > 0 such that the IVPassociated to the k-gKdV equation (1.3) with datau0 has solution

u C([0, T] : S(R)),satisfying

|u(x, t)| d ex3/2/3, x >1, t [0, T],for some constantd >0.

In the case of the free Schrodinger group{eit : t R}

eitu0(x) = (ei||2tu0)(x) = ei||2/4t

(4it)n/2 u0(x),

the fundamental solution does not decay. However, one has the identity

(1.11)

u(x, t) =eitu0(x) =

Rn

ei|xy|2/4t

(4it)n/2 u0(y) dy

= ei|x|

2/4t

(4it)n/2

Rn

e2ixy/4tei|y|2/4tu0(y) dy

= ei|x|

2/4t

(2it)n/2(ei||2/4tu0)

x2t

,

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where

f() = (2)n/2

Rn

eixf(x)dx.

Hence,cte

i|x|2/4t u(x, t) = (ei||2/4tu0)x

2t

, ct = (2it)

n/2,

which tells us thatei|x|2/4t u(x, t) is a multiple of the rescaled Fourier transform of

ei|y|2/4tu0(y). Thus, as we pointed out earlier, the behavior of the solution of the

free Schrodinger equation is closely related to uncertainty principles for the Fouriertransform. We shall study these uncertainty principles and their relation with theuniqueness properties of the solution of the Schrodinger equation (1.6). In the early1930s N. Wieners remark (see[28], [32], and [46]):

a pair of transforms f andg (f) cannot both be very small,motivated the works of G. H. Hardy [28], G. W. Morgan [46], and A. E. Ingham[32], which will be considered in detail in this note. However, before that we shallreturn to a review of some previous results concerning uniqueness properties ofsolutions of the Schrodinger equation which we mentioned earlier and which werenot motivated by the formula (1.11).

For solutions u(x, t) of the 1-D cubic Schrodinger equation (1.5) B. Y. Zhang[56]showed,

If u(x, t) = 0 for (x, t) (, a) {0, 1} (or (x, t) (a, ) {0, 1}) for some a R, thenu 0.

As was mentioned before, his proof is based on the inverse scattering method,which uses the fact that the equation in (1.5) is a completely integrable model.In [39] it was proved under general assumptions on F in (1.8)that:

Ifu1, u2 C([0, 1] :Hs(Rn)), with s >max{n/2; 2} are solutionsof the equation (1.6) withF as in (1.8) such that

u1(x, t) =u2(x, t), (x, t) cx0 {0, 1},wherecx0 denotes the complement of a conex0 with vertexx0Rn and opening0 such that if

(1.12) V : Rn [0, 1] C, with VL1tLx 0,andu C([0, 1] : L2(Rn)) is a strong solution of the IVP

(1.13)

tu= i( + V(x, t))u +G(x, t),

u(x, 0) = u0(x),

with

(1.14) u0, u1 u( , 1) L2(e2xdx), G L1([0, 1] :L2(e2xdx)),

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for some Rn, then there existscn independent of such that

(1.15)

sup0t1

exu( , t)L2(Rn)

cnexu0L2(Rn)+ exu1L2(Rn)+ 10

exG(, t)L2(Rn)dt.Notice that in the above result one assumes the existence of a reference L2-

solutionu of the equation (1.13) and then under the hypotheses (1.12) and (1.14)shows that the exponential decay in the time interval [0, 1] is preserved.

The estimate (1.15) can be combined with the subordination formula

(1.16) e|x|p/p

Rn

e1/px||q/q ||n(q2)/2 d, x Rn and p >1,

to get that for any >0 and a >1

(1.17)sup0t1 e

|x

|a

u( , t)L2

(Rn

)

cne|x|au0L2(Rn)+e|x|

a

u1L2(Rn)+ 1

0

e|x|a G(, t)L2(Rn)dt

.

Under appropriate assumptions on the potentialV(x, t) in(1.7), one writes

V(x, t)u= RV(x, t)u + (1 R)V(x, t)u= V(x, t)u +G(x, t),with R C0 , R(x) = 1,|x| < R, supported in|x| < 2R, and applies theestimate (1.17) by fixing R sufficiently large. Also under appropriate hypothesisonF andu, a similar argument can be used for the semilinear equation in (1.8).

The estimate (1.17) gives a control on the decay of the solution in the whole

time interval in terms of that at the endpoints and that of the external force. Aswe shall see below, a key idea will be to get improvements of this estimate basedon logarithmically convex versions of it.

We recall that if one considers the equation (1.6) with initial data u0 S(Rn)and a smooth potentialV(x, t) in (1.7) or smooth nonlinearityF in (1.8), it followsthat the corresponding solution satisfies that u C([T, T] : S(Rn)). This can beproved using the commutative property of the operators

L= t i, and j =xj+ 2txj , j= 1, . . . , n;see[29][30]. From the proof of this fact one also has that the persistence propertyof the solution u = u(x, t) (i.e., if the data u0

X, a function space, then the

corresponding solutionu() describes a continuous curve in X,u C([T, T] :X),T > 0) with data u0 L2(|x|m) can only hold if u0 Hs(Rn) with s 2m.Roughly speaking, for exponential weights one has a more involved argument wherethe time direction plays a role. Considering the IVP for the one-dimensional freeSchrodinger equation

(1.18)

tu= i

2xu, x, t R,

u(x, 0) =u0(x) L2(R),and assuming that exu0 L2(R), >0, then one formally has that

v(x, t) =exu(x, t)

satisfies the equationtv= i(x )2v.

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Thus,

v(x, 1) =exu(x, 1) L2(R) if e2 exu0 L2(R).However, if we knew that exu(x, 1), exu(x, 1) L2(R) integrating forwardin time the positive frequencies of e

x

u(x, t) and backward in time the negativefrequencies ofexu(x, t) one gets an estimate similar to that in (1.15) with = and G = 0. This argument motivates the idea behind Lemma 1and its proof.

The rest of this paper is organized as follows: Section 2 contains the resultsrelated to Hardys uncertainty principle including a short discussion on the ver-sion of this principle in terms of the heat flow. Section 3 contains those concernedwith Morgans uncertainty principle. In section 4 we shall consider the limitingcase in section 3. Also, section 4 includes the statements of some related forth-coming results. Earlier in the introduction we have discussed uniqueness resultsobtained under the assumption that the solution vanishes at two different times ina semispace (see[56], [33],[34],[20]). In section 2 similar uniqueness results will beestablished under a Gaussian decay hypothesis, in the whole space. In section 5 weshall obtain a unifying result, i.e., a uniqueness result under Gaussian decay in asemispace of Rn at two different times. The appendix contains an abstract lemmaand a corollary which will be used in the previous sections.

2. Hardys uncertainty principle

In [28]G. H. Hardy proved the following one-dimensional (n= 1) result:

If f(x) = O(e|x|2/2),

f() = O(e4||

2/2), and 1/ > 1/4,then f 0. Also, if 1/ = 1/4, f(x) is a constant multiple ofe|x|

2

/

2

.To our knowledge the available proofs of this result and its variants use complex

analysis, mainly appropriate versions of the PhragmenLindelof principle. Therehas also been considerable interest in a better understanding of this result and onextensions of it to other settings; see[5], [6], [12],[31], and[53]. In particular, theextension of Hardys result to higher dimensions n 2 (via Radon transform) wasgiven in [53].

The formula(1.11) allows us to rewrite this uncertainty principle in terms of thesolution of the IVP for the free Schrodinger equation

tu= iu, (x, t) Rn

(0, +

),

u(x, 0) =u0(x),

in the following manner:

Ifu(x, 0) =O(e|x|2/2), u(x, T) =O(e|x|

2/2), andT / > 1/4,thenu 0. Also, ifT/= 1/4, u has as initial datau0 equal toa constant multiple ofe(1/

2+i/4T)|y|2 .

The corresponding L2-version of Hardys uncertainty principle was established in[13]:

If e|x|2/2f, e4||

2/2

fare inL2(Rn)and1/ 1/4, thenf 0.

In terms of the solution of the Schrodinger equation it states:If e|x|

2/2u(x, 0), e||2/2u(x, T) are in L2(Rn) and T/ 1/4,

thenu 0.

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More generally, it was shown in [13] that:

If e|x|2/2f Lp(Rn), e4||2/2

f Lq(Rn), p, q [1, ] with at

least one of them finite and1/ 1/4, thenf 0.In[20] we proved a uniqueness result for solutions of (1.6) withFas in(1.7) for

bounded potentials V verifying that either

V(x, t) =V1(x) + V2(x, t)

withV1 real valued and

sup[0,T]

eT2|x|2/(t+(Tt))2V2(t)L(Rn)1/2 andVsatisfying one of the above conditions, is the zero solution.Notice that this result differs by a factor of 1/2 from that for the solution of thefree Schrodinger equation given by theL2-version of the Hardy uncertainty principledescribed above (T/ 1/4).

In[22] we showed that the optimal version of Hardys uncertainty principle in

terms ofL2-norms, as established in[13], holds for solutions of

(2.3) tu= i (u + V(x, t)u) , (x, t) Rn [0, T],such that (2.2) holds with T/ > 1/4 and for many general bounded poten-tialsV(x, t), while it fails for some complex-valued potentials in the endpoint case,T/= 1/4.

Theorem 3. LetuC([0, T]) : L2(Rn)) be a solution of equation (2.3). If thereexist positive constants andsuch thatT/ >1/4, and

e|x|

2/2u(0)

L2(Rn),

e|x|

2/2u(T)

L2(Rn)0.

The main idea consists of showing that either u 0 or there is a functionR: [1, 1] [0, 1] such that

(2.14) eR|x|2

4(1+R2t2) u(t)L2(Rn) e|x|2

u(1)R(t)L2(Rn)e|x|2

u(1)1R(t)L2(Rn) ,whereRis the smallest root of the equation

= R

4 (1 + R2) .

This gives the optimal improvement of the Gaussian decay of a free wave verifying(2.13), and we also see that if >1/8, thenu is zero.

The proof of these facts relies on new logarithmic convexity properties of freewaves verifying (2.13) and on those already established in[20]. In [20,Theorem 3],the positivity of the space-time commutator of the symmetric and skew-symmetricparts of the operator,

e|x|2

(t i) e|x|2

,

is used to prove thate|x|2

u(t)L2(Rn) is logarithmically convex in [1, 1]. Moreprecisely, definingf(x, t) =e|x|

2

u(x, t) =eitu0(x),

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it follows that

e|x|2

(t i) u= e|x|2 (t i) (e|x|2f) =tf Sf Af,where S is symmetric and A skew-symmetric with

S= i(4 x + 2n), A= i( + 42 |x|2),so that

[S;A] = 8( I) + 162 |x|2.Formally, using the abstract Lemma 3 (see the appendix) and the Heisenberg

inequality

f2L2(Rn) 2

n |x|fL2(Rn) fL2(Rn),

whose proof follows by integration by parts, one sees that

H(t) = f(t)2L2(Rn) = e|x|2

u(t)L2(Rn)is logarithmically convex so

e|x|2u(t)L2(Rn) e|x|2

u(1)1t2

L2(Rn)e|x|2

u(1)1+t2

L2(Rn),

when1 t 1.Settinga1 , we begin an iterative process, where at the kth step, we have k

smooth even functions, aj : [1, 1] (0, +), 1 j k, such that a1< a2 < < ak (1, 1),

F(ai)> 0, aj(1) =, j = 1, . . . , k ,

where

F(a) =

1

a a 3a2

2a + 32a3

and functions j : [1, 1] [0, 1], 1 j k, such that for t [.1, 1](2.15) eaj(t)|x|2u(t)L2(Rn) e|x|

2

u(1)j(t)L2(Rn)e|x|2

u(1)1j(t)L2(Rn) .These estimates follow from the construction of the functions ai, while the

method strongly relies on the following formal convexity properties of free waves:

(2.16) t

1

atlog Hb

2b

2||2F(a)

,

(2.17) t 1a tH a Rn ea|x|2|u|2 + |x|2|u|2 dx,where

Hb(t) = ea(t)|x+b(t)|2u(t)2L2(Rn) , H(t) = ea(t)|x|2

u(t)2L2(Rn), Rn anda, b: [1, 1] R are smooth functions with

a >0, F(a)> 0 in [1, 1].Once the kth step is completed, we take a = ak in (2.16) with a certain choice

ofb= bk, verifying b(1) =b(1) = 0, and then a certain test is performed. Whenthe answer to the test is positive, it follows thatu 0. Otherwise, the logarithmicconvexity associated to (2.16) allows us to find a new smooth function ak+1 in

[1, 1] witha1< a2< < ak < ak+1, (1, 1),

and verifying the same properties as a1, . . . , ak.

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When the process is infinite, we have (2.15) for all k 1 and there are twopossibilities:

either limk+

ak(0) = + or limk+

ak(0)< +.

In the first case and (2.15) one has that u0, while in the second, the sequenceak is shown to converge to an even function a verifying

(2.18)

a 3a22a + 32a3 = 0, [1, 1],a(1) =.

Because

a(t) = R

4 (1 + R2t2), R R+,

are all the possible even solutions of this equation, a must be one of them, and

=

R

4 (1 + R2)

for some R >0. In particular, u 0, when >1/8.As was already mentioned above, our proof of Theorem 3(the case of nonzero

potentialsV =V(x, t)) is based on the extension of the above convexity propertiesto the nonfree case.

Theorem 4 establishes the sharpness of the result in Theorem 3 by giving anexample of a complex valued potential V(x, t) verifying (2.1) and a nontrivial so-lution u C([0, T] : L2(Rn)) of (2.3) for which (2.2) holds with T/ = 1/4.Thus, one may ask, Is it possible to construct a real-valued potential V(x, t) ver-ifying the same properties, i.e., satisfying (2.1) and having a nontrivial solution

u C([0, T] : L2(Rn)) of (2.3) such that (2.2) holds with T/= 1/4 ?The same question concerning the sharpness of the above result presents itself

in the case of time-independent potentials V = V(x). In this regard, we considerthe stationary problem

(2.19) w+ V(x)w= 0, x Rn, V L(Rn),and recall V. Z. Meshkovs result in [45]:

Ifw H2loc(Rn) is a solution of (2.19) such that

(2.20)

Rnea|x|

4/3 |w(x)|2dx < , a >0,

then u 0.Moreover, it was also proved in [45]that for complex potentials V, the exponent

4/3 in (2.20) is optimal. However, it has been conjectured that for real-valuedpotentials the optimal exponent should be 1 (see also [7]for a quantitative form ofthese results and applications to Anderson localization of Bernoulli models).

More generally, it was established in [23](see also [14]):

If w H2loc(Rn) is a solution of (2.19) with a complex-valued po-tentialV satisfying

V(x) =V1(x) + V2(x)

such that

(2.21) |V1(x)| c1(1 + |x|2)/2 , [0, 1/2),

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andV2 is real valued supported in{x :|x| 1} such that(rV2(x))< c2|x|2 , a

= min{a; 0},

then there existsa= a(V; c1; c2; )> 0 such that if(2.22)

Rn

ea|x|r |w(x)|2dx < , r= (4 2)/3,

then u 0.In addition, one can take the value r = 1 in (2.20) by assuming > 1/2 in

(2.21).It was also proved in[14]that for complex potentials, these results for [0, 1/2)

are sharp.By noticing that given a solution (x) of the eigenvalue problem

(2.23) +V(x)=, x Rn,with R,ifV(x) =V(x) + satisfies the hypothesis of the previous result since

u(x, t) =eit (x),

solves the evolution equation

(2.24) tu= i(u +V(x)u), x Rn, t R,then one gets a lower bound for the value of the strongest possible decay rate ofnontrivial solutions u(x, t) of (2.24) at two different times.

As a direct consequence of Theorem 3, we have the following application con-cerning the uniqueness of solutions for semilinear equations of the form (1.6) with

Fas in (1.8).

Theorem 6. Let u1 and u2 be strong solutions in C([0, T], Hk(Rn)), k > n/2 of

the equation (1.6) with F as in (1.8) such that F Ck and F(0) = uF(0) =uF(0) = 0. If there are andpositive withT/ >1/4 such that

e|x|2/2 (u1(0) u2(0)), e|x|2/2 (u1(T) u2(T)) L2(Rn),

thenu1 u2.In Theorem6we did not attempt to optimize the regularity assumption on the

solutions u1, u2.

By fixing u2 0 Theorem6provides a restriction on the possible decay at twodifferent times of a nontrivial solution u1 of equation (1.6) withFas in (1.8). It isan open question to determine the optimality of this kind of result. More precisely,for the standard semilinear Schrodinger equations

(2.25) tu= i(u + |u|1u), >1,one has the standing wave solutions

u(x, t) =e t(x), >0,

whereis the unique (up to translation) positive solution of the elliptic problem

+ =

|

|1,

which has a linear exponential decay, i.e.,

(x) =O(ec|x|), as|x| ,

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for an appropriate value ofc >0 (see[54], [3],[4], and[42]). Whether or not thesestanding waves are the solutions of (2.25) having the strongest possible decay attwo different times is an open question.

Hardys uncertainty principle also admits a formulation in terms of the heat

equationtu= u, t >0, x Rn,

whose solution with data u(x, 0) = u0(x) can be written as

u(x, t) =etu0(x) =

Rn

e|xy|2/4t

(4t)n/2 u0(y) dy.

More precisely, Hardys uncertainty principle can restated in the following equiv-alent forms:

(i) If u0 L2(Rn) and there exists T > 0 such that e|x|2/(2T) eTu0 L2(Rn) for some 2, then u0 0.

(ii) If u0 S(Rn

) (tempered distribution) and there exists T > 0 such thate|x|

2/(2T) eTu0 L(Rn) for some < 2, then u0 0. Moreover, if= 2, then u0 is a constant multiple of the Dirac delta measure.

In fact, applying Hardys uncertainty principle to eTu0, one has that e|x|2

2TeTu0and eT||

2 eTu0 =u0 in L2(Rn) with 2 4 implies eu0 0. Then, backwarduniqueness arguments (see for example [44, Chapter 3, Theorem 11]) show thatu0 0.

In [20] we proved the following weaker extension of this result for parabolicoperators with lower order variable coefficients:

Theorem 7. LetuC([0, 1] :L

2

(Rn

)) L2

([0, T] :H

1

(Rn

)) be a solution of theIVP tu= u + V(x, t)u, inRn (0, 1],u(x, 0) =u0(x),

whereV L(Rn [0, 1]).

If

u0 and e|x|2

2 u(1) L2(Rn),for some 0, with

ab > cosp 2 ,

thenf 0.

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In [31]Beurling and Hormander showed:

Iff L1(R) and

(3.1) R R |f(x)||f()|e|x | dx d < ,thenf 0.

This result was extended to higher dimensions n 2 in[6]and [48]:Iff L2(Rn), n 2 and

(3.2)

Rn

Rn

|f(x)||f()|e|x | dxd < ,then 0.

We observe that from (3.1) and (3.2) it follows that:

Ifp

(1, 2], 1/p + 1/q= 1, a, b >0, and

(3.3)Rn

|f(x)| eap|x|p

p dx +Rn

|f()| e bq||qq d < ,thenab 1 f 0.

Notice that in the case p = q = 2 this gives us an L1-version of Hardys un-certainty result discussed above, and for p < 2 an n-dimensional L1-version ofMorgans uncertainty principle.

In the one-dimensional case (n= 1), the optimal L1-version of Morgans resultin (3.3),

(3.4) R |f(x)| eap|x|p

p dx + R |f()| ebq||q

q d cosp 2 f 0.was established in [6] and [2] (for further results, see [5] and references therein).A sharp condition for a, b, p in (3.4) in higher dimension seems to be unknown.However, in [6]it was shown:

If f L2(Rn), 1 < p 2 and 1/p+ 1/q = 1 are such that forsomej = 1, . . . , n,

(3.5)

Rn

|f(x)|eap|xj |

p

p dx < +Rn

|f()|e bq|j |qq d < .Ifab > cos p 2 , then f 0.Ifab cosp 2 if n= 1, and ab >1 if n 2,

thenu0 0.

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Related with Morgans uncertainty principle, one has the following result dueto Gelfand and Shilov. In [26]they considered the class Zpp , p >1, defined as thespace of all functions(z1, . . . , zn) which are analytic for all values ofz1, . . . , zn Cand such that

|(z1, . . . , zn)| C0 enj=1 jCj |zj |p ,where the Cj, j = 0, 1, . . . , nare positive constants and j = 1 for zj nonreal andj =1 for zj real, j = 1, . . . , n, and showed that the Fourier transform of thefunction space Zpp is the space Z

qq , with 1/p + 1/q= 1.

Notice that the class Zpp with p2 is closed with respect to multiplication byeic|x|

2

. Thus, ifu0 Zpp , p 2, then by (1.11)one has that|eitu0(x)| d(t) ea(t)|x|

q

,

for some functions d, a : R (0, ).In [21]the following results were established:

Theorem 8. Given p (1, 2) there exists Mp > 0 such that for any solutionu C([0, 1] : L2(Rn)) of

tu= i (u + V(x, t)u) , in Rn [0, 1],withV =V(x, t) complex valued, bounded (i.e.,VL(Rn[0,1]) C) and(3.7) lim

R+VL1([0,1]:L(Rn\BR))= 0,

satisfying that for some constants a0, a1, a2> 0

(3.8) Rn |u(x, 0)|2 e2a0|x|

p

dx Mp,

then u 0.Corollary 1. Givenp (1, 2) there existsNp > 0 such that ifuC([0, 1] :L2(Rn))

is a solution of tu= i(u + V(x, t)u),

withV =V(x, t) complex valued, bounded (i.e.,VL(Rn[0,1]) C) and

limR

10

sup|x|>R

|V(x, t)|dt= 0,

and there exist, >0 such that

(3.11)

Rn

|u(x, 0)|2e2 p |x|p/pdx +Rn

|u(x, 1)|2e2 q |x|q/qdx < ,

1/p + 1/q= 1, with

(3.12) > Np,

then u 0.

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As a consequence of Corollary1, one obtains the following result concerning theuniqueness of solutions for the semilinear equations (1.6) withFas in (1.8)

(3.13) itu + u= F(u, u).

Theorem 9. Given p (1, 2) there exists Np>0 such that ifu1, u2 C([0, 1] : Hk(Rn))

are strong solutions of (3.13) with k Z+, k > n/2, F : C2 C, F Ck, andF(0) =uF(0) =uF(0) = 0, and there exist, >0 such that

(3.14) ep |x|p/p (u1(0) u2(0)) , e

q |x|q/q (u1(1) u2(1)) L2(Rn),1/p + 1/q= 1, with

(3.15) > Np,

thenu1 u2.Notice that the conditions (3.10)and (3.12) are independent of the size of the

potential and there is not any a priori regularity assumption on the potential V(x, t).The result in [6] (see (3.5)) can be extended to our setting with a nonoptimal

constant. More precisely,

Corollary 2. The conclusions in Corollary 1 still hold with a different constantNp >0 if one replaces the hypothesis (3.11) by the following one-dimensional ver-sion

(3.16)

Rn

|u(x, 0)|2e2 p |xj |p/pdx < +Rn

|u(x, 1)|2e2 q |xj |q/qdx < ,

for somej= 1, . . . , n.Similarly, the nonlinear version of Theorem9 still holds, with different constant

Np > 0, if one replaces the hypothesis (3.14) by

ep |xj |p/p (u1(0) u2(0)) , e

q |xj |q/q (u1(1) u2(1)) L2(Rn),for j = 1, . . . , n.

In [21]we did not attempt to give an estimate of the universal constant Np.The limiting case p= 1 will be considered in the next section.The main idea in the proof of these results is to combine an upper estimate with

a lower one to obtain the desired result. The upper estimate is based on the decayhypothesis on the solution at two different times (see Lemma 1). In previous workswe had been able to establish these estimates from assumptions that at time t= 0and t = 1 involving the same weight. However, in our case (Corollary1) we havedifferent weights at time t = 0 and t = 1. To overcome this difficulty, we carryout the details with the weight eaj |x|

p

, 1 < p < 2, j = 0 at t = 0 and j = 1 att = 1, with a0 fixed and a1 = k Z+ as in (3.9). Although the powers |x|p inthe exponential are equal at time t = 0 and t = 1 to apply our estimate (Lemma1) we also need to have the same constant in front of them. To achieve this weapply the conformal or Appell transformation described above, to get solutions andpotentials, whose bounds depend onk Z+. Thus we have to consider a family ofsolutions and obtain estimates on their asymptotic value as k .

The proof of the lower estimate is based on the positivity of the commutatoroperator obtained by conjugating the equation with the appropriate exponentialweight (see Lemma3in the appendix).

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4. PaleyWiener theorem and uncertainty principleof Ingham type

This section is concerned with the limiting case p= 1 in the previous section.

It is easy to see that iff L1

(Rn

) is nonzero and has compact support, thenfcannot satisfy a condition of the typef(y) =O(e|y|) for any >0. However, itmay be possible to havef L1(Rn) a nonzero function with compact support, suchthatf() = O(e(y)|y|),(y) being a positive function tending to zero as|y| .

In the one-dimensional case (n= 1) soon after Hardys result described above,A. E. Ingham [32]proved the following:

There existsf L1(R)nonzero, even, vanishing outside an intervalsuch thatf(y) = O(e(y)|y|) with (y) being a positive functiontending to zero at infinity if and only if

(y)

y dy < .In a similar direction the PaleyWiener theorem[50] gives a characterization of

a function or distribution with compact support in term of analyticity propertiesof its Fourier transform.

Regarding our results discussed above it would be interesting to identify a classof potentials V(x, t) for which a result of the following kind holds:

Ifu C([0, 1] : L2(Rn)) is a nontrivial solution of the IVP

(4.1)

tu= i(u + V(x, t)u), (x, t) Rn [0, 1],u(x, 0) =u0(x),

with u0 L2(Rn) having compact support, then e|x| u(, t) /L2(Rn) for any >0 and any t (0, 1].

In this direction we have the following result which will appear in[40]:

Theorem 10. Assume thatu C([0, 1] :L2(Rn)) is a strong solution of the IVP(2.4) with

suppu0 BR(0) = {x Rn :|x| R},(4.2) Rn

e2a1|x| |u(x, 1)|2 dx < , a1> 0,(4.3)

and

(4.4) VL(Rn[0,1]) = M0,with

(4.5) limR+

VL1([0,1]:L(Rn\BR))= 0.

Then, there existsb= b(n)> 0 (depending only on the dimensionn) such that if

a1R (M0+ 1)

b,

then u

0.

A similar question can be raised for results of the type described above due toA. E. Ingham in[32] and possible extensions to higher dimensions n 2.

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It would be interesting to obtain extensions of the above results characterizingthe decay of the solutionu(x, t) to the equation(1.6) withFas in (1.8) associatedto datau0 L2(Rn) with compact support or withu0 C0 (Rn). In this direction,some results can be deduced as a consequence of Theorem10;see [40].

5. Hardys uncertainty principle in a half-space

In the introduction we have briefly reviewed some uniqueness results establishedfor solutions of the Schrodinger equation vanishing at two different times in a semis-pace of Rn (see[56], [15],[33], [34],[20]). In section 2, we have studied uniquenessresults obtained under the hypothesis that the solution of the Schrodinger equationat two different times has an appropriate Gaussian decay in the whole space Rn.In this section, we shall deduce a unified result, i.e., a uniqueness result under thehypothesis that at two different times the solution of the Schrodinger equation hasGaussian decay in just a semispace of Rn.

Theorem 11. Assume thatu C([0, 1] :L2((0, ) Rn1)) is a strong solutionof the IVP

(5.1)

tu= i( + V(x, t))u,

u(x, 0) =u0(x),

with 10

3/21/2

|x1u(x, t)|2 dxdt < ,(5.2)

V : Rn [0, 1] C, V L(Rn [0, 1]),(5.3)

and

(5.4) limR+

10

V(t)L({x1>R}) dt= 0.

Assume that

(5.5)

x1>0

ec0 |x1|2 |u(x, 0)|2 dx < ,

x1>0

ec1 |x1|2 |u(x, 1)|2 dx < ,

withc0, c1> 0 sufficiently large. Then u 0.Remarks. (a) Note that in Theorem11,the solution does not need to be defined forx1 0. In this sense, this is a stronger result that the uniqueness results in [56],[39],[33], [34], and[15], which required that the solution be defined in Rn [0, 1]and beC([0, 1] :L2(Rn)).

On the other hand, we need to assume the condition (5.2). Note that[39]alsoneeds an extra assumption on u, stronger that (5.2), but that in [33], whichamong other things removed any extra assumption on u, but still required thesolution to be defined in Rn [0, 1] and be inC([0, 1] :L2(Rn)). If, in the setting ofTheorem11,we know that u is a solution in Rn [0, 1] and is inC([0, 1] : L2(Rn)),then we can dispose the hypothesis (5.2)as follows.

First, as in the first step of the proof of Theorem 11, we can use the Appelltransformation to reduce to the casec1 = c2= 2. Then, using (x1) aregularized

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convex function which agrees with x+1 for x1 > 1 , x1 2 t(1 t)|u(x, t)|2e2(x

+1)

2

dxdt 0, we are in the situation of Theorem 11, and hence u 0 on{x1 > 2} [0, 1]. Finally, Izakovs result in [35] concludes that u 0 (moreprecisely, the version of Izakovs result proved in [33], which does not require uto exist for1< x1 0

}remains invariant.Step 2. Upper bounds.

We definev(x, t) =(x1) u(x, t),

with C(R), nondecreasing with (x1) 1 if x1 > 3/2, and (x1) 0 ifx1< 1/2. Therefore,

(5.6) tv= i v+ i V(x, t)v+ i F(x, t), F(x, t) = 2 x1u (x1) + u (x1).

Using (5.2), we can apply Lemma1to get that

(5.7)

sup0t1

ex1v( , t)L2(Rn)

cnex1v(0)L2(Rn)+ ex1v(1)L2(Rn)

+

10

ex1 F(, t)L2(Rn)dt+ 1

0

ex1 V {x1

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Thus, from the formula (1.16) (withp = 2 and n = 1) and (5.8) we obtain that

sup0t1

e|x1|2v( , t)L2(Rn)

e|x1|2v(0)L2(Rn)+ e|x1|2v(1)L2(Rn)+ c +V ec R2.Thus,

(5.9) sup0t1

e|x1|2v( , t)L2(Rn) c.

Combining this and the equation for v, we shall get a smoothing estimate. Usingthe notation

H(t) = f2L2(Rn) = f2,with

f(x, t) =e|x1|2

v(x, t)

and the abstract Lemma3(see the appendix), one formally has that

(5.10)2t H 2tRe(tf Sf Af, f)

+ 2 (Stf+ [S,A] f, f) + e|x1|2

(F+ V v)2,with

e|x1|2

(t i )(e|x1|2

f) =tf Sf Af=e|x1|2

(F+ V v),

where S= i(4x1 x1+2) is symmetric, A= i(+4x21) is skew-symmetric, andF is as in (5.6). Since

[S;A] =

82x1

+ 162 x21.

using the inequalityRn

(|x1f|2 + 42|x1|2|f|2) dx=Rn

e2 |x1|2

(|x1u|2 2|u|2)dx

2 Rn

|f|2 dx,

together with Corollary3we conclude that

(5.11)

10

t(1 t) |x1v(x, t)|2 e2 |x1|

2

e2 |x1|2

dxdt c.

Combining and (5.9) and (5.11) one gets that

(5.12)

sup0t1

e|x1|2v( , t)L2(Rn)

+

10

t(1 t)|x1v(x, t)|2 e2 |x1|

2

e2 |x1|2 |dxdt c.

Step 3. We recall the following result, which is a slight variation of that proven indetail in[16] (Lemma 3.1, page 1818):

Lemma 2. Assume that R > 0 and : [0, 1] R is a smooth function. Then,there exists c= c(n;

+

)> 0 such that the inequality

(5.13) 3/2

R2

e| x1x01R +(t)|2gL2(dxdt)

c e| x1x01R +(t)|2(it+ )g

L2(dxdt)

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holds when > cR2 and g C0 (Rn+1) is supported in the set{(x, t) = (x1, . . . , xn, t) Rn+1 :| x1 x01

R + (t)| 1}.

Now, we will choose x01 =R/2, 0 (t) a, with a= 3/2 1/R, (t) =a,on 3/8 t 5/8, (t) = 0, for t [0, 1/4] [3/4, 1], and R C(R) withR(x1) = 1 on 1< x1 < R 1, and R(x1) = 0 for x1< 1/2 or x1> R.

Also we choose C(R) with (x1) = 0, x1 1 and (x1) = 1, x11 + 1/2R.

We notice that, up to translation, we can assume that

(5.14)

5/83/8

2

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But, if 1/2< x1

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and

N(t) (Stf+ [S,A] f, f) /H tf Af Sf2/ (2H) .Moreover, if

(6.2) |tf Af Sf| M1|f| + G, inRn

[0, 1], St+ [S,A] M0,and

M2= sup[0,1]

G(t)/f(t)

is finite, then log H(t) is logarithmically convex in [0, 1] and there is a universalconstantNsuch that

(6.3) H(t) eN(M0+M1+M2+M21+M22)H(0)1tH(1)t, when0 t 1.By multiplying the formula (6.1) by t(1 t), integrating the result over [0, 1],

and using integration by parts, one gets the following smoothinginequality:

Corollary 3. With the same hypotheses and notation as in Lemma 3

(6.4)

2

10

t(1 t) (Stf+ [S,A] f, f) dt + 1

0

H(t) dt H(0) + H(1)

+ 2

10

(1 2t)Re(tf Sf Af, f)dt

+

10

t(1 t)tf Af Sf22 dt.

About the authors

Luis Escauriaza is a professor in the Department of Mathematics at the Univer-sity of the Basque Country in Bilbao, Spain.

Carlos E. Kenig is the Louis Block Distinguished Service Professor in Mathemat-ics and the College at the University of Chicago. He has been awarded the SalemPrize (1984) and the AMS Bocher Prize (2008), and he was a plenary speaker atthe International Congress of Mathematicians in Hyderabad in 2010.

Gustavo Ponce is a professor of mathematics at the University of California,Santa Barbara. He was an invited speaker at the International Congress of Math-ematicians in Berlin in 1998.

Luis Vega is a professor in the Department of Mathematics at the University of

the Basque Country in Bilbao, Spain. He was an invited speaker at the InternationalCongress of Mathematicians in Madrid in 2006.

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UPV/EHU, Depto. de Matematicas, Apto. 644, 48080 Bilbao, SpainE-mail address: [email protected]

Department of Mathematics, University of Chicago, Chicago, Illinois 60637E-mail address: [email protected]

Department of Mathematics, University of California, Santa Barbara, California93106

E-mail address: [email protected]

UPV/EHU, Depto. de Matematicas, Apto. 644, 48080 Bilbao, SpainE-mail address: [email protected]

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