100 years of Weyl’s lawweyl.math.toronto.edu/victor_ivrii_Publications/preprints/Talk_9.pdfGöttingen, math.-phys. Kl., Sitzung vom 18. Mai 1912. **) Für Parallelepipede mit irrationalem

100 years of Weyl’s lawMeeting of AMS,

Pullman, WA, April 22–23, 2017

Victor Ivrii

Department of Mathematics, University of Toronto

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 1 / 39

Table of Contents

Table of Contents

1 Origin

2 Sharper remainder estimates

3 Generalizations

4 Variants


Origin

In 1911 a 26-years old mathemati-cian, a former student of DavidHilbert, Hermann Weyl publisheda very important paper [W1] Überdie asymptotische Verteilung derEigenwerte (About the asymptoticdistribution of eigenvalues),

whichfollowed by four more papers [W2,W3] (1912), [W4] (1913) Über dieRandwertaufgabe der Strahlungs-theorie und asymptotische Spek-tralgesetze (About the boundaryvalue problem of the theory ofradiation and asymptotic spectrallaws), [W5] (1915) and he re-turned to this topic much later in[W7] in 1950.

Figure: Hermann Weyl


Origin

In 1911 a 26-years old mathemati-cian, a former student of DavidHilbert, Hermann Weyl publisheda very important paper [W1] Überdie asymptotische Verteilung derEigenwerte (About the asymptoticdistribution of eigenvalues), whichfollowed by four more papers [W2,W3] (1912), [W4] (1913) Über dieRandwertaufgabe der Strahlungs-theorie und asymptotische Spek-tralgesetze (About the boundaryvalue problem of the theory ofradiation and asymptotic spectrallaws), [W5] (1915) and he re-turned to this topic much later in[W7] in 1950.

Figure: Hermann Weyl


Ueber die asymptotische Verteilung der Eigenwerte.

Von

Hermann W eyl, Göttingen.

Vorgelegt durch Herrn D. Hilbertin der Sitzung vom 25. Februar 1911.

Im folgenden teile ich einige einfache Sätze über die Eigen-werte von Integralgleichungen mit, welche namentlich deren asymp-totische Verteilung betreffen. Die Anwendung der gewonnenen Resultate auf die Differentialgleichung .du + A.u = 0 (Satz X) liefert insbesondere die Lösung eines Problems, auf dessen Wich-tigkeit neuerdings A. Sommerfeld (auf der Naturforscherver-sammlung zu Königsberg 1)) und H. A. Loren t z (in seinen hier in Göttingen zu Beginn dieses Semesters gehaltenen Vorträgen 2)) nachdrücklich hingewiesen haben.

Die Eigenwerte eines symmetrischen Kernes K ( s, t) - nur um solche Kerne handelt es sich im folgenden -bezeichne ich, indem ich sie nach der Größe ihres absoluten} Betrages anordne, mit

_!._ , _!._, ... ; in dieser Reihe soll natürlich jeder Eigenwert so " "2 oft vertreten sein, als seine Vielfachheit angibt. Die reziproken positiven Eigenwerte, gleichfalls nach ihrer Größe angeordnet,

+ + - -~ heißen u1 , u2 , • • • , die negativen " 1 , "~ , • • • • In entsprechender Weise verwende ich u', u" u. s. w. zur Bezeichnung der reziproken Eigenwerte anderer Kerne K', K" u. s. w.

Meine Untersuchungen basieren auf dem folgenden

1) Physikalische Zeitschrift, Bd. XI (1910), S. 1061. 2) Physikalische Zeitschrift, Bd. XI (1910), S. 1248.

Origin

For Dirichlet Laplacian in a bounded domain Ω H.Weyl (1911) proved that

N(λ) = c0 mes(Ω)λd2 + o(λ

d2 ) (1)

as λ→ +∞ and

later conjectured [W4] (1913) that

N(λ) = c0 mes(Ω)λd2 ∓ c1 mesd−1(∂Ω)λ

d−12 + o(λ

d−12 ). (2)

Definition

Here N(λ) is a number of eigenvalues of Laplacian −∆, which are lesserthan λ.

This formula (1) was actually conjectured independently byArnold Sommerfeld and Hendrik Lorentz in 1910 who stated the Weyl’sLaw as a conjecture based on the book of Lord Rayleigh “The Theory ofSound” (1887).


https://en.wikipedia.org/wiki/Arnold_Sommerfeldhttps://en.wikipedia.org/wiki/Hendrik_Lorentzhttps://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleighhttps://archive.org/details/theoryofsound02raylrichhttps://archive.org/details/theoryofsound02raylrich

Origin

For Dirichlet Laplacian in a bounded domain Ω H.Weyl (1911) proved that

N(λ) = c0 mes(Ω)λd2 + o(λ

d2 ) (1)

as λ→ +∞ and later conjectured [W4] (1913) that

N(λ) = c0 mes(Ω)λd2 ∓ c1 mesd−1(∂Ω)λ

d−12 + o(λ

d−12 ). (2)

Definition

Here N(λ) is a number of eigenvalues of Laplacian −∆, which are lesserthan λ.

This formula (1) was actually conjectured independently byArnold Sommerfeld and Hendrik Lorentz in 1910 who stated the Weyl’sLaw as a conjecture based on the book of Lord Rayleigh “The Theory ofSound” (1887).


https://en.wikipedia.org/wiki/Arnold_Sommerfeldhttps://en.wikipedia.org/wiki/Hendrik_Lorentzhttps://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleighhttps://archive.org/details/theoryofsound02raylrichhttps://archive.org/details/theoryofsound02raylrich

Weyl, asymptotische Spektralgesetze. 199

haltene Zahl, so stimmt die Summe rechterhand in der ersten UngleichungVlenbis auf einen Fehler < Const. —fr- mit= n/"

B · n-2/*überein:

a„ > 4- n'/3 l - Const. -r- .= B L Vü JEine obere Grenze für o„ ist gegeben durch

a» < i» < i ™'/3 1 + Const.B L

Der Versuch, diese Abschätzungen wesentlich weiter zu treiben, etwa

neben dem ersten Gliede -=· n/3 noch das 2. Glied einer vielleicht existierenden,D

nach ansteigenden Potenzen von n fortschreitenden „asymptotischen Reihe" zuermitteln, scheint gegenwärtig wenig aussichtsreich. Wenn J ein Würfelvon der Kantenlänge l ist, (der Fall, der immer die Grundlage bildet)kann freilich aus neueren zahlentheoretischen Untersuchungen der HerrenVorono'i, Sierpinski, Landau*), bei denen sehr schwierige und subtile Hülfs-mittel zur Verwendung kommen, das zweite Glied einer solchen asym-ptotischen Entwicklung entnommen werden; man bekommt hier für xn (umnur von dem leichteren Membranproblem zu sprechen)

(6 2 )2/3 + f · (6 2 )1/8

mit einer Abweichung < Const. n/6+s (s irgendeine feste positive Zahl).Der genaue Fehler, ich meine die Differenz der rechten und linken Seitein der letzten asymptotischen Gleichung, ist wahrscheinlich eine zahlen-theoretische Funktion von höchst unregelmäßigem Verhalten, die sichasymptotisch nicht mehr mit einer Potenz von n vergleichen läßt**). Manwird geneigt sein, die Schuld daran den Kanten und Ecken des Würfelszuzuschreiben und bei Räumen z. B., die von regulär-analytischen Flächen

*) Vorono'i, dieses Journal Bd. 126 (1903), S. 241—282; Sierpinsla, Pracematematyczno-fizyczne, Bd. 17 (1906), S. 77—118; Landau, Nachr. d. Ges. d. Wiss.,Göttingen, math.-phys. Kl., Sitzung vom 18. Mai 1912.

**) Für Parallelepipede mit irrationalem Kantenverhältnis scheint bei demheutigen Stande der Zahlentheorie sogar die Ermittlung des zweiten Gliedes schonnicht mehr möglich zu sein.

Origin

To explain (1) consider a rectangular box of the size a1 × a2 × · · · × ad .Then N(λ) equals to the number of integer points in the domain

Θ ={

(m1, . . . ,md) ∈ Z+ d ,m21a21

+ . . .+m2da2d

<λ

π2}. (3)

which should be approximately equal to the volume of the domain Θ

(which is 2−d -th of ellipsoid with semiaxis π−1λ12 a1, . . . , π

−1λ12 ad) i.e.

ωd(2π)−1λ

12 a1 · · · (2π)−1λ

12 ad = ωd(2π)

−dλd2 mes(Ω),

mes(Ω) = a1 · · · ad ,

which would be exactly the first term in (1). Here and below ωd is avolume of the unit ball in Rd .


Origin

Weyl conjecture (2) was the result of a more precise analysis of the sameproblem: for Dirichlet boundary condition we do not count points withmi = 0 (blue) and for Neumann problem we count them, so in (2) will be“−” for Dirichlet, and “+” for Neumann, c1 = 14 (2π)

1−dωd−1.


Origin

The proof of (1) by Weyl was based on this formula for boxes andvariational arguments he invented. Let us cover domain by small boxes:

Inner and boundary boxes


Origin

Note that N(λ) = max dim L , where L runs over all subspaces of H onwhich quadratic form ‖∇u‖2 − λ‖u‖2 is negative and H is Sobolev spaceH1(Ω) in the case of Neumann boundary problem andH10 (Ω) = {u ∈ H1(Ω), u|∂Ω = 0} in the case of Dirichlet boundaryproblem.

Then for Dirichlet boundary problem first we tighten conditions to Lrequiring that u = 0 in all boundary boxes and on all walls between boxes;then N(λ) decreases and

N(λ) ≥ Nnew(λ) =∑ι

Nι,D(λ) ≥∑ι

c0 mesBι λd/2 − o(λd/2) ≥ c0(mesX − �)λd/2 − o(λd/2)

with arbitrarily small � > 0; here ι runs inner boxes only.


Origin

Note that N(λ) = max dim L , where L runs over all subspaces of H onwhich quadratic form ‖∇u‖2 − λ‖u‖2 is negative and H is Sobolev spaceH1(Ω) in the case of Neumann boundary problem andH10 (Ω) = {u ∈ H1(Ω), u|∂Ω = 0} in the case of Dirichlet boundaryproblem.Then for Dirichlet boundary problem first we tighten conditions to Lrequiring that u = 0 in all boundary boxes and on all walls between boxes;then N(λ) decreases and

N(λ) ≥ Nnew(λ) =∑ι

Nι,D(λ) ≥∑ι

c0 mesBι λd/2 − o(λd/2) ≥ c0(mesX − �)λd/2 − o(λd/2)

with arbitrarily small � > 0; here ι runs inner boxes only.


Origin

On the other hand, let us loosen conditions to L , allowing u be differentfrom 0 everywhere on boundary boxes and also allowing to bediscontinuous on the boxes walls; then N(λ) increases and

N(λ) ≤ Nnew(λ) =∑ι

Nι,N(λ) ≤∑ι

c0 mesBι λd/2 + o(λd/2) ≤ c0(mes X̄ + �)λd/2 + o(λd/2)

with arbitrarily small � > 0; here ι runs inner and boundary boxes and weassume that mes(∂X ) = 0.

Combining these two inequalities we get (1).

More delicate arguments allow us to drop this condition mes(∂X ) = 0.


Origin

On the other hand, let us loosen conditions to L , allowing u be differentfrom 0 everywhere on boundary boxes and also allowing to bediscontinuous on the boxes walls; then N(λ) increases and

N(λ) ≤ Nnew(λ) =∑ι

Nι,N(λ) ≤∑ι

c0 mesBι λd/2 + o(λd/2) ≤ c0(mes X̄ + �)λd/2 + o(λd/2)

with arbitrarily small � > 0; here ι runs inner and boundary boxes and weassume that mes(∂X ) = 0. Combining these two inequalities we get (1).

More delicate arguments allow us to drop this condition mes(∂X ) = 0.


Sharper remainder estimates


Actually Weyl invented this method (called Dirichlet–Neumannbracketing) only in 1912.

Richard Courant, in [Cour] (1920), pushing this method to its limits

proved remainder estimate to O(λd−1

2 log λ) for bounded domains withC∞ boundary.

Actually both H. Weyl and R. Courant considered only d = 2, 3. ProbablyTorsten Carleman was the first to consider arbitrary d ≥ 2. He alsoinvented Tauberian methods [C1, C2] (1935, 1936), based on the analysisof

Tr f (H, t) =

∫f (λ, t) dλN(λ). (4)



The method of hyperbolic operator (which is a variant of the Tauberian

method1) allowed to improve remainder estimates to O(λd−1

2 ) (Boris

Levitan [Lev1] (1952), and V. Avakumovič [Av] (1956)) and to o(λd−1

2 )(J. J. Duistermaat–Victor Guillemin [DG] (1975)–for manifolds withoutboundary,

and to O(λd−1

2 ) (Richard Seeley, [See1, See2] (1978)), and to o(λd−1

2 )(Victor Ivrii, [Ivr1] (1980))–for manifolds with the boundaries.

1) With f (λ, t) = e iλt or f (λ, t) = cos(√λt) etc.



To have sharp remainder estimate o(λd−1

2 ) one needs

Geometrical condition

The set of all periodic geodesics (geodesic billiards–for manifolds with theboundary) has measure 0.

While there is a conjecture

Ivrii’ conjecture

In the Euclidean domain the set of all periodic geodesic billiards hasmeasure 0

it turned out to be one of the most difficult problems in the theory ofmathematical billiards.

It has been proven for generic domains and for some special domains.

Under stronger condition remainder estimate O(λd−1

2 /| log λ|) or evenO(λ

d−12−δ) has been recovered (f.e. for polyhedral Euclidean domains).


Generalizations

Generalizations: Neumann Laplacian

What about Neumann boundary condition? Neumann Laplacian could bea very different beast. In fact, even in the bounded domain its spectrumcould be essential in which case N(λ) =∞ (if there is an essentialspectrum below λ).

However for bounded domain Weyl asymptotic (1) holds provided thefollowing condition is satisfied:

Cone condition

Each point of x ∈ Ω can be touched by the vertex of the cone of theheight h and angle α, located inside Ω (we are allowed to move and rotatecone but h > 0 and α > 0 must be fixed (albeit arbitrarily small).

This condition provides H1(Ω)→ H1(Rd) continuation.


Generalizations

Cone condition holds Cone condition fails


Generalizations

Domains with cusps

xf (x1)

Here f (x1)→ 0 as x1 → +∞ and cross-section is f (x1)G .


Generalizations

Weyl law understood literally claims that N(λ) =∞ if mes(Ω) =∞.

Butit is not the case for Dirichlet problem: N(λ) λ−

12 }) (5)

because too narrow (less than �λ−12 ) and infinitely long cusp cannot host

eigenvalues less than λ.

But for Neumann Laplacian situation is drastically different: essentialspectrum appears even if f (x) = e−cx ! Only for f (x) = O(e−x

α) with

α > 1 all spectrum is discrete.


Generalizations

Weyl law understood literally claims that N(λ) =∞ if mes(Ω) =∞. Butit is not the case for Dirichlet problem: N(λ) λ−

12 }) (5)




α) with



Generalizations


12 }) (5)



But for Neumann Laplacian situation is drastically different: essentialspectrum appears even if f (x) = e−cx !

Only for f (x) = O(e−xα

) withα > 1 all spectrum is discrete.


Generalizations


12 }) (5)




α) with



Generalizations

Remark

Even if Weyl’s law understood literally fails in the examples above, theanswer is often given in the terms of Weyl’s law for some other operators,

f.e. Schrödingier operators defined on the base of the cusp but may bewith a potential which is a function which values are operators in theauxiliary space H, which is L 2(Ω) for Dirichlet Laplacian in the domainwith a thick cusp, and just C for a Neumann Laplacian in the domain withan ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variablesx ′ 7→ x ′/f (x1) then operator −∆ becomes (roughly) such Schrödingeroperator−∂21 − f (x1)−2∆G and separation of variables leads to−∂21 + f (x1)−2µm where µm are eigenvalues of the Laplacian in G . It isfine for Dirichlet Laplacian (µm > 0) but for Neumann Laplacian we get aproblem as µ1 = 0.

However more educated change of variables (taking in account themeasure) leads to an extra potential 14 (∂1 log f )

2 which saves the day!


Generalizations

Remark

Even if Weyl’s law understood literally fails in the examples above, theanswer is often given in the terms of Weyl’s law for some other operators,f.e. Schrödingier operators defined on the base of the cusp but may bewith a potential which is a function which values are operators in theauxiliary space H,

which is L 2(Ω) for Dirichlet Laplacian in the domainwith a thick cusp, and just C for a Neumann Laplacian in the domain withan ultra-thin cusp.





Generalizations

Remark

Even if Weyl’s law understood literally fails in the examples above, theanswer is often given in the terms of Weyl’s law for some other operators,f.e. Schrödingier operators defined on the base of the cusp but may bewith a potential which is a function which values are operators in theauxiliary space H, which is L 2(Ω) for Dirichlet Laplacian in the domainwith a thick cusp,

and just C for a Neumann Laplacian in the domain withan ultra-thin cusp.





Generalizations

Remark

Even if Weyl’s law understood literally fails in the examples above, theanswer is often given in the terms of Weyl’s law for some other operators,f.e. Schrödingier operators defined on the base of the cusp but may bewith a potential which is a function which values are operators in theauxiliary space H, which is L 2(Ω) for Dirichlet Laplacian in the domainwith a thick cusp, and just C for a Neumann Laplacian in the domain withan ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variablesx ′ 7→ x ′/f (x1) then operator −∆ becomes (roughly) such Schrödingeroperator−∂21 − f (x1)−2∆G and separation of variables leads to−∂21 + f (x1)−2µm where µm are eigenvalues of the Laplacian in G . It isfine for Dirichlet Laplacian (µm > 0)

but for Neumann Laplacian we get aproblem as µ1 = 0.




Generalizations

Remark

Even if Weyl’s law understood literally fails in the examples above, theanswer is often given in the terms of Weyl’s law for some other operators,f.e. Schrödingier operators defined on the base of the cusp but may bewith a potential which is a function which values are operators in theauxiliary space H, which is L 2(Ω) for Dirichlet Laplacian in the domainwith a thick cusp, and just C for a Neumann Laplacian in the domain withan ultra-thin cusp.





Generalizations

Variational methods

Variational methods introducedby Weyl were developed by manyauthors. I want to mention onlyMikhail Birman (and his school),Elliott Lieb (and his school) andBarry Simon (and his school).

Mikhail Birman


Generalizations

Generalized Weyl law

N− ≈ (2π)−dωd∫

Vd2− dx (6)

where N− is a number of negative eigenvalues of −∆ + V ,V− = max(−V , 0).

In particular, for semiclassical asymptotics for operator −h2∆ + V (x) ash→ +0 we expect

N−h ∼ (2πh)−dωd

∫V

d2− dx (7)


Generalizations

Estimates

The famous Cwikel-Lieb-Rozenblum estimate for number of negativeeigenvalues of Schrd̈ingier −∆ + V (x) operator in Rd (or in domain withDirichlet boundary conditions:

N− ≤ C∫

Vd2− dx (8)

with C = Cd depending only on d ≥ 3.

Grisha Rozenblum, [Roz1] (1972), proved actually much more generalestimate. Other proofs (based on different ideas) were given by MichaelCwikel (1977), Elliott Lieb (1980), Peter Li and Shing-Tung Yau (1983),Joseph Conlon (1985).

Bad news: Cd in CLR does not coincide (and cannot coincide) withsemiclassical constant in (6)!


Generalizations

Estimates


N− ≤ C∫

Vd2− dx (8)


Grisha Rozenblum, [Roz1] (1972), proved actually much more generalestimate.

Other proofs (based on different ideas) were given by MichaelCwikel (1977), Elliott Lieb (1980), Peter Li and Shing-Tung Yau (1983),Joseph Conlon (1985).



Generalizations

Estimates


N− ≤ C∫

Vd2− dx (8)


Grisha Rozenblum, [Roz1] (1972), proved actually much more generalestimate. Other proofs (based on different ideas) were given by MichaelCwikel (1977), Elliott Lieb (1980), Peter Li and Shing-Tung Yau (1983),Joseph Conlon (1985).



Generalizations

CLR however could be used in conjugation with variational and Tauberianmethods to prove asymptotics, including those with sharp remainderestimates. CLR is used to prove that certain “singular” zones providesmall contribution to eigenvalue counting function.

CLR implies Lieb–Thirring inequality∑λj

Variants

Local Weyl Law

Local Weyl Law:∫e(x , x , λ)ψ(x) dx = Tr E (λ)ψ ≈ (2π)−dωd

∫V

d2−ψ(x) dx (10)

where e(x , y , λ) is the Schwartz’ kernel of the spectral projector E (λ) ofoperator H = −∆ + V and ψ(x) is a C∞0 cut-off function.

If ψ = 1, we get N(λ) in the right-hand expression, but generally we wantto assemble 1 from differently scaled ψι.


Variants

Pointwise Weyl Law

Pointwise Weyl Law (needs to be justified under appropriate assumptions)claims:

e(x , x , 0) ≈ (2π)−dωdVd2− (x) (11)

where e(x , y , λ) is the Schwartz kernel of the spectral projector ofH = −∆ + V (x).

If negative spectrum of H is purely point, then

e(x , y , 0) =∑λj

Variants

Application: Thomas-Fermi Theory

Consider a large (heavy) atom or molecule; it is described by QuantumHamiltonian

HN =∑

1≤j≤N−∆xj −

∑1≤m≤M,1≤j≤N

Zm|xj − ym|

︸︷︷︸V (x)

+∑

1≤j

Variants

If electrons were not interacting between themselves but the field potentialwas −W (x) then they would occupy lowest eigenvalues and ground statewave functions would be (anti-symmetrized) φ1(x1)φ2(x2) . . . φN(xN)where φj and λj are eigenfunctions and eigenvalues of H = −∆−W (x).

Then the local electron density would be ρΨ =∑

1≤j≤N |φj(x)|2 andaccording to pointwise Weyl law

ρΨ(x) ≈1

6π2(W + ν)

32+ (14)

where ν = λN .

This density would generate potential −|x |−1 ∗ ρΨ and we would haveW ≈ V − |x |−1 ∗ ρΨ.


Variants



1≤j≤N |φj(x)|2

andaccording to pointwise Weyl law

ρΨ(x) ≈1

6π2(W + ν)

32+ (14)

where ν = λN .



Variants



1≤j≤N |φj(x)|2 andaccording to pointwise Weyl law

ρΨ(x) ≈1

6π2(W + ν)

32+ (14)

where ν = λN .



Variants

Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:

V −W TF = |x |−1 ∗ ρTF, (15)

ρTF =1

6π2(W TF + ν)

32+, (16)∫

ρTF dx = N (17)

where ν ≤ 0 is called chemical potential and in fact approximates λN .

Thomas-Fermi theory has been rigorously justified (with pretty good errorestimates).


Variants

Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:

V −W TF = |x |−1 ∗ ρTF, (15)

ρTF =1

6π2(W TF + ν)

32+, (16)∫

ρTF dx = N (17)

where ν ≤ 0 is called chemical potential and in fact approximates λN .Thomas-Fermi theory has been rigorously justified (with pretty good errorestimates).


Variants

In fact the ground state energy is given by

EN =(6π2)

53

15π2

∫ (ρTF

53 − V ρTF

)dx

− 12

xρTF(x)ρTF(y)|x − y |−1 dxdy + O(N2) (18)

and justified Scott correction term ∼ N2 and Dirac and Schwingercorrection terms ∼ N

53 (so the error is O(N

53−δ) with some δ > 0).

More details (including models with magnetic field – either external orself-generated) are in my talk

http://weyl.math.toronto.edu/victor2/preprints/Talk 10.pdf


http://weyl.math.toronto.edu/victor2/preprints/Talk_10.pdf

Variants

Going to subspaces

In many problems of mathematical physics the domain of operator is densenot in L 2(Ω,Rm) but in some its subspace.

For example, if domain has certain symmetries we can consider subspacesof functions having “similar” symmetries. This could be crucial: forexample, Neumann Laplacian in the domain with a cusp (which is notultra-thin) has an essential spectrum. However, if this domain issymmetric with respect to the plane passing through the middle of thecusp, and we consider only functions which are odd, the essential spectrumdisappears and operator has discrete spectrum.

Another example: in the simplest problems arising in electromagneticoscillations and in hydrodynamics we need to consider −∆ on the space ofsolenoidal fields: ∇ · u = 0. Crystal optics provides more complicatedexamples.


Variants

Going to subspaces



Another example: in the simplest problems arising in electromagneticoscillations and in hydrodynamics we need to consider −∆ on the space ofsolenoidal fields: ∇ · u = 0.

Crystal optics provides more complicatedexamples.


Variants

Going to subspaces



Another example: in the simplest problems arising in electromagneticoscillations and in hydrodynamics we need to consider −∆ on the space ofsolenoidal fields: ∇ · u = 0. Crystal optics provides more complicatedexamples.


Variants

Integrated density of states

There are problems when spectrum is essential. What global quantityreplacing N(λ) we can study?

Assume that V is periodic or quasi periodic (whatever it means) anddomain is Rd . Let Ω be a box or a ball centred at origin, considerstretched domain LΩ and consider either

N(λ) = limL→∞

1

mes(LΩ)

∫LΩ

e(x , x , λ) dx (19)

or

N(λ) = limL→∞

1

mes(LΩ)NLΩ(λ) (20)

where in the former case e(x , y , λ) is defined for Rd but integrated overLΩ and in the letter NLΩ(λ) =

∫LΩ eLΩ(x , x , λ) dx is calculated for LΩ.

Usually these definitions are equivalent.

N(λ) defined this way is called integrated density of states.


Variants

Integrated density of states

There are problems when spectrum is essential. What global quantityreplacing N(λ) we can study?

Assume that V is periodic or quasi periodic (whatever it means) anddomain is Rd . Let Ω be a box or a ball centred at origin, considerstretched domain LΩ and consider either

N(λ) = limL→∞

1

mes(LΩ)

∫LΩ

e(x , x , λ) dx (19)

or

N(λ) = limL→∞

1

mes(LΩ)NLΩ(λ) (20)

where in the former case e(x , y , λ) is defined for Rd but integrated overLΩ and in the letter NLΩ(λ) =

∫LΩ eLΩ(x , x , λ) dx is calculated for LΩ.

Usually these definitions are equivalent.

N(λ) defined this way is called integrated density of states.Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 31 / 39

Variants

Spectral shift function

Another object appears in the Scattering Theory when two operators Hand H0 are close at infinity. Then one can study Krein-Birman spectralshift function ∫ (

eH(x , x , λ)− eH0(x , x , λ))dx (21)

which its not the difference of two integrals as both of them diverge.


Variants

Reference I

W. Arendt W., R. Nittka R., W. Peter W. andF. Steiner. Weyls Law: Spectral Properties of the Laplacian inMathematics and Physics, pp. 1–71, in Mathematical Analysis ofEvolution, Information, and Complexity, by W. Arendt and W.P.Schleich, Wiley-VCH, 2009.

V. G. Avakumovič. Über die eigenfunktionen auf geschlossenriemannschen mannigfaltigkeiten. Math. Z., 65:324–344 (1956).

T. Carleman. Propriétes asymptotiques des fonctionsfondamentales des membranes vibrantes. In C. R. 8-ème Congr. Math.Scand., Stockholm, 1934, pages 34–44, Lund (1935).

T. Carleman. Über die asymptotische Verteilung der Eigenwertepartieller Differentialgleichungen.Ber. Sachs. Acad. Wiss. Leipzig, 88:119–132 (1936).


Variants

Reference II

R. Courant.Über die Eigenwerte bei den Differentialgleichungen dermathematischen Physik. Mat. Z., 7:1–57 (1920).

J. J. Duistermaat and V. Guillemin. The spectrum of positiveelliptic operators and periodic bicharacteristics. Invent. Math.,29(1):37–79 (1975).

J. J. Duistermaat and V. Guillemin. The spectrum of positiveelliptic operators and periodic bicharacteristics. Invent. Math.,29(1):37–79 (1975).

L. Hörmander. The spectral function of an elliptic operator. ActaMath., 121:193–218 (1968).


Variants

Reference III

L. Hörmander. On the Riesz means of spectral functions andeigenfunction expansions for elliptic differential operators. In YeshivaUniv. Conf., November 1966, volume 2 of Ann. Sci. Conf. Proc.,pages 155–202. Belfer Graduate School of Sci. (1969).

V. Ivrii. Second term of the spectral asymptotic expansion for theLaplace-Beltrami operator on manifold with boundary. Funct. Anal.Appl., 14(2):98–106 (1980).

V. Ivrii. Accurate spectral asymptotics for elliptic operators that actin vector bundles. Funct. Anal. Appl., 16(2):101–108 (1982).

V. Ivrii. Microlocal Analysis and Precise Spectral Asymptotics,Springer-Verlag, SMM, 1998, xv+731.


Variants

Reference IV

V. Ivrii. Microlocal Analysis and Sharp Spectral Asymptotics, inprogress: available online athttp://www.math.toronto.edu/ivrii/monsterbook.pdf

V. Ivrii. 100 years of Weyl’s law, Bulletin of Mathematical Sciences,Springer (2016).

B. M. Levitan. On the asymptotic behaviour of the spectralfunction of the second order elliptic equation. Izv. AN SSSR, Ser.Mat., 16(1):325–352 (1952) (in Russian).

B. M. Levitan. Asymptotic behaviour of the spectral function ofelliptic operator. Russian Math. Surveys, 26(6):165–232 (1971).

S. Nonnenmacher, Counting stationary modes: a discrete view ofgeometry and dynamics, Talk at WEYL LAW at 100, a workshop atthe Fields Institute, September 19–21, (2012).


http://www.math.toronto.edu/ivrii/monsterbook.pdfhttp://link.springer.com/article/10.1007/s13373-016-0089-yhttps://www.fields.utoronto.ca/programs/scientific/12-13/public_lectures/Nonnenmacher.pdfhttps://www.fields.utoronto.ca/programs/scientific/12-13/public_lectures/Nonnenmacher.pdf

Variants

Reference V

G. Rozenblioum. The distribution of the discrete spectrum ofsingular differential operators. English transl.: Sov. Math., Izv. VUZ20(1):63-71 (1976).

R. Seeley. A sharp asymptotic estimate for the eigenvalues of theLaplacian in a domain of R3. Advances in Math., 102(3):244–264(1978).

R. Seeley. An estimate near the boundary for the spectral functionof the Laplace operator. Amer. J. Math., 102(3):869–902 (1980).

H. Weyl. Über die Asymptotische Verteilung der Eigenwerte. Nachr.Konigl. Ges. Wiss. Göttingen, 110–117 (1911).

H. Weyl. Das asymptotische Verteilungsgesetz linearen partiellenDifferentialgleichungen. Math. Ann., 71:441–479 (1912).


Variants

Reference VI

H. Weyl. Über die Abhängigkeit der Eigenschwingungen einerMembran von deren Begrenzung. J. Für die Angew. Math., 141:1–11(1912).

H. Weyl. Über die Randwertaufgabe der Strahlungstheorie undasymptotische Spektralgeometrie. J. Reine Angew. Math., 143,177-202 (1913).

H. Weyl. Das asymptotische Verteilungsgesetz derEigenschwingungen eines beliebig gestalteten elastischen Körpers.Rend. Circ. Mat. Palermo. 39:1–49 (1915).

H. Weyl. Quantenmechanik und Gruppentheorie, Zeitschrift fürPhysik, 46:1–46 (1927) (see The Theory of Groups and QuantumMechanics, Dover, 1950, xxiv+422).


Variants

Reference VII

H. Weyl. Ramifications, old and new, of the eigenvalue problem.Bull. Amer. Math. Soc. 56(2):115–139 (1950).


OriginSharper remainder estimatesGeneralizationsVariants

Documents

100 years of Weyl’s lawweyl.math.toronto.edu/victor_ivrii_Publications/preprints/Talk_9.pdfGöttingen, math.-phys. Kl., Sitzung vom 18. Mai 1912. **) Für Parallelepipede mit irrationalem