100 years of Weyl’s law Meeting 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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 2 / 39 which followed by four more papers [W2, W3] (1912), [W4] (1913) Uber¨ die Randwertaufgabe der Strahlungs- theorie und asymptotische Spek- tralgesetze (About the boundary value problem of the theory of radiation and asymptotic spectral laws), [W5] (1915) and he re- turned to this topic much later in [W7] in 1950.

Origin In 1911 a 26-years old mathemati- cian, a former student of David Hilbert, Hermann Weyl published a very important paper [W1] Uber¨ die asymptotische Verteilung der Eigenwerte (About the asymptotic distribution of eigenvalues),

Figure: Hermann Weyl

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 3 / 39 Origin In 1911 a 26-years old mathemati- cian, a former student of David Hilbert, Hermann Weyl published a very important paper [W1] Uber¨ die asymptotische Verteilung der Eigenwerte (About the asymptotic distribution of eigenvalues), which followed by four more papers [W2, W3] (1912), [W4] (1913) Uber¨ die Randwertaufgabe der Strahlungs- theorie und asymptotische Spek- tralgesetze (About the boundary value problem of the theory of radiation and asymptotic spectral laws), [W5] (1915) and he re- turned to this topic much later in Figure: Hermann Weyl [W7] in 1950.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 3 / 39 Ueber die asymptotische Verteilung der Eigenwerte.

Von

Hermann Weyl, 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 1 sammlung zu Königsberg )) 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. later conjectured [W4] (1913) that

d d−1 d−1 N(λ) = c0 mes(Ω)λ 2 ∓ c1 mesd−1(∂Ω)λ 2 + o(λ 2 ). (2)

Definition Here N(λ) is a number of eigenvalues of Laplacian −∆, which are lesser than λ. This formula (1) was actually conjectured independently by Arnold Sommerfeld and Hendrik Lorentz in 1910 who stated the Weyl’s Law as a conjecture based on the book of Lord Rayleigh“The Theory of Sound” (1887).

Origin

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

d d N(λ) = c0 mes(Ω)λ 2 + o(λ 2 ) (1) as λ → +∞ and

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 5 / 39 This formula (1) was actually conjectured independently by Arnold Sommerfeld and Hendrik Lorentz in 1910 who stated the Weyl’s Law as a conjecture based on the book of Lord Rayleigh“The Theory of Sound” (1887).

Origin

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

d d N(λ) = c0 mes(Ω)λ 2 + o(λ 2 ) (1) as λ → +∞ and later conjectured [W4] (1913) that

d d−1 d−1 N(λ) = c0 mes(Ω)λ 2 ∓ c1 mesd−1(∂Ω)λ 2 + o(λ 2 ). (2)

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 5 / 39 Origin

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

d d N(λ) = c0 mes(Ω)λ 2 + o(λ 2 ) (1) as λ → +∞ and later conjectured [W4] (1913) that

d d−1 d−1 N(λ) = c0 mes(Ω)λ 2 ∓ c1 mesd−1(∂Ω)λ 2 + o(λ 2 ). (2)

Definition Here N(λ) is a number of eigenvalues of Laplacian −∆, which are lesser than λ. This formula (1) was actually conjectured independently by Arnold Sommerfeld and Hendrik Lorentz in 1910 who stated the Weyl’s Law as a conjecture based on the book of Lord Rayleigh“The Theory of Sound” (1887).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 5 / 39 Weyl, asymptotische Spektralgesetze. 199 haltene Zahl, so stimmt die Summe rechterhand in der ersten Ungleichung Vlen bis auf einen Fehler < Const. —fr- mit = n/" 2 B · n- /* überein: /3 a„ => B4- n' l - Const. -r- . L Vü J Eine obere Grenze für o„ ist gegeben durch /3 1 a» < i» < Bi ™' + Const. L Der Versuch, diese Abschätzungen wesentlich weiter zu treiben, etwa /3 neben dem ersten Gliede -=· n noch das 2. Glied einer vielleicht existierenden, D nach ansteigenden Potenzen von n fortschreitenden „asymptotischen Reihe" zu ermitteln, scheint gegenwärtig wenig aussichtsreich. Wenn J ein Würfel von der Kantenlänge l ist, (der Fall, der immer die Grundlage bildet) kann freilich aus neueren zahlentheoretischen Untersuchungen der Herren Vorono'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 (um nur von dem leichteren Membranproblem zu sprechen) 2 2/3 2 1/8 <-> (6 ) + f · (6 ) /6+s mit einer Abweichung < Const. n (s irgendeine feste positive Zahl). Der genaue Fehler, ich meine die Differenz der rechten und linken Seite in der letzten asymptotischen Gleichung, ist wahrscheinlich eine zahlen- theoretische Funktion von höchst unregelmäßigem Verhalten, die sich asymptotisch nicht mehr mit einer Potenz von n vergleichen läßt**). Man wird geneigt sein, die Schuld daran den Kanten und Ecken des Würfels zuzuschreiben und bei Räumen z. B., die von regulär-analytischen Flächen

*) Vorono'i, dieses Journal Bd. 126 (1903), S. 241—282; Sierpinsla, Prace matematyczno-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 dem heutigen Stande der Zahlentheorie sogar die Ermittlung des zweiten Gliedes schon nicht mehr möglich zu sein. which should be approximately equal to the volume of the domain Θ −d −1 1 −1 1 (which is 2 -th of ellipsoid with semiaxis π λ 2 a1, . . . , π λ 2 ad ) i.e.

−1 1 −1 1 −d d ωd (2π) λ 2 a1 ··· (2π) λ 2 ad = ωd (2π) λ 2 mes(Ω),

mes(Ω) = a1 ··· ad ,

which would be exactly the first term in (1). Here and below ωd is a d volume of the unit ball in R .

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

2 2  + d m1 md λ Θ = (m1,..., md ) ∈ Z , 2 + ... + 2 < 2 . (3) a1 ad π

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 7 / 39 which would be exactly the first term in (1). Here and below ωd is a d volume of the unit ball in R .

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

2 2  + d m1 md λ Θ = (m1,..., md ) ∈ Z , 2 + ... + 2 < 2 . (3) a1 ad π which should be approximately equal to the volume of the domain Θ −d −1 1 −1 1 (which is 2 -th of ellipsoid with semiaxis π λ 2 a1, . . . , π λ 2 ad ) i.e.

−1 1 −1 1 −d d ωd (2π) λ 2 a1 ··· (2π) λ 2 ad = ωd (2π) λ 2 mes(Ω),

mes(Ω) = a1 ··· ad ,

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 7 / 39 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

2 2  + d m1 md λ Θ = (m1,..., md ) ∈ Z , 2 + ... + 2 < 2 . (3) a1 ad π which should be approximately equal to the volume of the domain Θ −d −1 1 −1 1 (which is 2 -th of ellipsoid with semiaxis π λ 2 a1, . . . , π λ 2 ad ) i.e.

−1 1 −1 1 −d d ωd (2π) λ 2 a1 ··· (2π) λ 2 ad = ωd (2π) λ 2 mes(Ω),

mes(Ω) = a1 ··· ad , which would be exactly the first term in (1). Here and below ωd is a d volume of the unit ball in R .

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 7 / 39 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

2 2  + d m1 md λ Θ = (m1,..., md ) ∈ Z , 2 + ... + 2 < 2 . (3) a1 ad π which should be approximately equal to the volume of the domain Θ −d −1 1 −1 1 (which is 2 -th of ellipsoid with semiaxis π λ 2 a1, . . . , π λ 2 ad ) i.e.

−1 1 −1 1 −d d ωd (2π) λ 2 a1 ··· (2π) λ 2 ad = ωd (2π) λ 2 mes(Ω),

mes(Ω) = a1 ··· ad , which would be exactly the first term in (1). Here and below ωd is a d volume of the unit ball in R .

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 7 / 39 Origin

Weyl conjecture (2) was the result of a more precise analysis of the same problem: for Dirichlet boundary condition we do not count points with mi = 0 (blue) and for Neumann problem we count them, so in (2) will be 1 1−d “−” for Dirichlet, and “+” for Neumann, c1 = 4 (2π) ωd−1.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 8 / 39 Origin The proof of (1) by Weyl was based on this formula for boxes and variational arguments he invented. Let us cover domain by small boxes:

Inner and boundary boxes

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 9 / 39 Then for Dirichlet boundary problem first we tighten conditions to L requiring that u = 0 in all boundary boxes and on all walls between boxes; then N(λ) decreases and

X N(λ) ≥ Nnew(λ) = Nι,D(λ) ≥ ι X d/2 d/2 d/2 d/2 c0 mes Bι λ − o(λ ) ≥ c0(mes X − )λ − o(λ ) ι 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 on which quadratic form k∇uk2 − λkuk2 is negative and H is Sobolev space H1(Ω) in the case of Neumann boundary problem and 1 1 H0 (Ω) = {u ∈ H (Ω), u|∂Ω = 0} in the case of Dirichlet boundary problem.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 10 / 39 Origin

Note that N(λ) = max dim L , where L runs over all subspaces of H on which quadratic form k∇uk2 − λkuk2 is negative and H is Sobolev space H1(Ω) in the case of Neumann boundary problem and 1 1 H0 (Ω) = {u ∈ H (Ω), u|∂Ω = 0} in the case of Dirichlet boundary problem. Then for Dirichlet boundary problem first we tighten conditions to L requiring that u = 0 in all boundary boxes and on all walls between boxes; then N(λ) decreases and

X N(λ) ≥ Nnew(λ) = Nι,D(λ) ≥ ι X d/2 d/2 d/2 d/2 c0 mes Bι λ − o(λ ) ≥ c0(mes X − )λ − o(λ ) ι with arbitrarily small  > 0; here ι runs inner boxes only.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 10 / 39 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 different from 0 everywhere on boundary boxes and also allowing to be discontinuous on the boxes walls; then N(λ) increases and

X N(λ) ≤ Nnew(λ) = Nι,N(λ) ≤ ι X d/2 d/2 d/2 d/2 c0 mes Bι λ + o(λ ) ≤ c0(mes X¯ + )λ + o(λ ) ι with arbitrarily small  > 0; here ι runs inner and boundary boxes and we assume that mes(∂X ) = 0.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 11 / 39 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 different from 0 everywhere on boundary boxes and also allowing to be discontinuous on the boxes walls; then N(λ) increases and

X N(λ) ≤ Nnew(λ) = Nι,N(λ) ≤ ι X d/2 d/2 d/2 d/2 c0 mes Bι λ + o(λ ) ≤ c0(mes X¯ + )λ + o(λ ) ι with arbitrarily small  > 0; here ι runs inner and boundary boxes and we assume that mes(∂X ) = 0. Combining these two inequalities we get (1).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 11 / 39 Origin

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

X N(λ) ≤ Nnew(λ) = Nι,N(λ) ≤ ι X d/2 d/2 d/2 d/2 c0 mes Bι λ + o(λ ) ≤ c0(mes X¯ + )λ + o(λ ) ι with arbitrarily small  > 0; here ι runs inner and boundary boxes and we assume that mes(∂X ) = 0. Combining these two inequalities we get (1). More delicate arguments allow us to drop this condition mes(∂X ) = 0.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 11 / 39 Richard Courant, in [Cour] (1920), pushing this method to its limits d−1 proved remainder estimate to O(λ 2 log λ) for bounded domains with C ∞ boundary. Actually both H. Weyl and R. Courant considered only d = 2, 3. Probably Torsten Carleman was the first to consider arbitrary d ≥ 2. He also invented Tauberian methods [C1, C2] (1935, 1936), based on the analysis of Z Tr f (H, t) = f (λ, t) dλN(λ). (4)

Sharper remainder estimates Sharper remainder estimates

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 12 / 39 Actually both H. Weyl and R. Courant considered only d = 2, 3. Probably Torsten Carleman was the first to consider arbitrary d ≥ 2. He also invented Tauberian methods [C1, C2] (1935, 1936), based on the analysis of Z Tr f (H, t) = f (λ, t) dλN(λ). (4)

Sharper remainder estimates Sharper remainder estimates

Actually Weyl invented this method (called Dirichlet–Neumann bracketing) only in 1912. Richard Courant, in [Cour] (1920), pushing this method to its limits d−1 proved remainder estimate to O(λ 2 log λ) for bounded domains with C ∞ boundary.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 12 / 39 Sharper remainder estimates Sharper remainder estimates

Actually Weyl invented this method (called Dirichlet–Neumann bracketing) only in 1912. Richard Courant, in [Cour] (1920), pushing this method to its limits d−1 proved remainder estimate to O(λ 2 log λ) for bounded domains with C ∞ boundary. Actually both H. Weyl and R. Courant considered only d = 2, 3. Probably Torsten Carleman was the first to consider arbitrary d ≥ 2. He also invented Tauberian methods [C1, C2] (1935, 1936), based on the analysis of Z Tr f (H, t) = f (λ, t) dλN(λ). (4)

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 12 / 39 d−1 d−1 and to O(λ 2 ) (Richard Seeley, [See1, See2] (1978)), and to o(λ 2 ) (Victor Ivrii, [Ivr1] (1980))–for manifolds with the boundaries.

Sharper remainder estimates

The method of hyperbolic operator (which is a variant of the Tauberian 1) d−1 method allowed to improve remainder estimates to O(λ 2 ) (Boris d−1 Levitan [Lev1] (1952), and V. Avakumoviˇc[Av] (1956)) and to o(λ 2 ) (J. J. Duistermaat–Victor Guillemin [DG] (1975)–for manifolds without boundary,

√ 1) With f (λ, t) = eiλt or f (λ, t) = cos( λt) etc. Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 13 / 39 Sharper remainder estimates

The method of hyperbolic operator (which is a variant of the Tauberian 1) d−1 method allowed to improve remainder estimates to O(λ 2 ) (Boris d−1 Levitan [Lev1] (1952), and V. Avakumoviˇc[Av] (1956)) and to o(λ 2 ) (J. J. Duistermaat–Victor Guillemin [DG] (1975)–for manifolds without boundary,

d−1 d−1 and to O(λ 2 ) (Richard Seeley, [See1, See2] (1978)), and to o(λ 2 ) (Victor Ivrii, [Ivr1] (1980))–for manifolds with the boundaries.

√ 1) With f (λ, t) = eiλt or f (λ, t) = cos( λt) etc. Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 13 / 39 While there is a conjecture Ivrii’ conjecture In the Euclidean domain the set of all periodic geodesic billiards has measure 0 it turned out to be one of the most difficult problems in the theory of mathematical billiards. It has been proven for generic domains and for some special domains.

d−1 Under stronger condition remainder estimate O(λ 2 /| log λ|) or even d−1 −δ O(λ 2 ) has been recovered (f.e. for polyhedral Euclidean domains).

Sharper remainder estimates

d−1 To have sharp remainder estimate o(λ 2 ) one needs Geometrical condition The set of all periodic geodesics (geodesic billiards–for manifolds with the boundary) has measure 0.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 14 / 39 It has been proven for generic domains and for some special domains.

d−1 Under stronger condition remainder estimate O(λ 2 /| log λ|) or even d−1 −δ O(λ 2 ) has been recovered (f.e. for polyhedral Euclidean domains).

Sharper remainder estimates

d−1 To have sharp remainder estimate o(λ 2 ) one needs Geometrical condition The set of all periodic geodesics (geodesic billiards–for manifolds with the boundary) has measure 0.

While there is a conjecture Ivrii’ conjecture In the Euclidean domain the set of all periodic geodesic billiards has measure 0 it turned out to be one of the most difficult problems in the theory of mathematical billiards.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 14 / 39 d−1 Under stronger condition remainder estimate O(λ 2 /| log λ|) or even d−1 −δ O(λ 2 ) has been recovered (f.e. for polyhedral Euclidean domains).

Sharper remainder estimates

d−1 To have sharp remainder estimate o(λ 2 ) one needs Geometrical condition The set of all periodic geodesics (geodesic billiards–for manifolds with the boundary) has measure 0.

While there is a conjecture Ivrii’ conjecture In the Euclidean domain the set of all periodic geodesic billiards has measure 0 it turned out to be one of the most difficult problems in the theory of mathematical billiards. It has been proven for generic domains and for some special domains.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 14 / 39 Sharper remainder estimates

d−1 To have sharp remainder estimate o(λ 2 ) one needs Geometrical condition The set of all periodic geodesics (geodesic billiards–for manifolds with the boundary) has measure 0.

While there is a conjecture Ivrii’ conjecture In the Euclidean domain the set of all periodic geodesic billiards has measure 0 it turned out to be one of the most difficult problems in the theory of mathematical billiards. It has been proven for generic domains and for some special domains.

d−1 Under stronger condition remainder estimate O(λ 2 /| log λ|) or even d−1 −δ O(λ 2 ) has been recovered (f.e. for polyhedral Euclidean domains).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 14 / 39 However for bounded domain Weyl asymptotic (1) holds provided the following condition is satisfied: Cone condition Each point of x ∈ Ω can be touched by the vertex of the cone of the height h and angle α, located inside Ω (we are allowed to move and rotate cone but h > 0 and α > 0 must be fixed (albeit arbitrarily small).

1 1 d This condition provides H (Ω) → H (R ) continuation.

Generalizations Generalizations: Neumann Laplacian

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 15 / 39 1 1 d This condition provides H (Ω) → H (R ) continuation.

Generalizations Generalizations: Neumann Laplacian

What about Neumann boundary condition? Neumann Laplacian could be a very different beast. In fact, even in the bounded domain its spectrum could be essential in which case N(λ) = ∞ (if there is an essential spectrum below λ). However for bounded domain Weyl asymptotic (1) holds provided the following condition is satisfied: Cone condition Each point of x ∈ Ω can be touched by the vertex of the cone of the height h and angle α, located inside Ω (we are allowed to move and rotate cone but h > 0 and α > 0 must be fixed (albeit arbitrarily small).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 15 / 39 Generalizations Generalizations: Neumann Laplacian

What about Neumann boundary condition? Neumann Laplacian could be a very different beast. In fact, even in the bounded domain its spectrum could be essential in which case N(λ) = ∞ (if there is an essential spectrum below λ). However for bounded domain Weyl asymptotic (1) holds provided the following condition is satisfied: Cone condition Each point of x ∈ Ω can be touched by the vertex of the cone of the height h and angle α, located inside Ω (we are allowed to move and rotate cone but h > 0 and α > 0 must be fixed (albeit arbitrarily small).

1 1 d This condition provides H (Ω) → H (R ) continuation.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 15 / 39 Generalizations

Cone condition holds Cone condition fails

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 16 / 39 Generalizations Domains with cusps

x f (x1)

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 17 / 39 But it is not the case for Dirichlet problem: N(λ) < ∞ provided f (x) → 0 as x → +∞. In fact,

d − 1 N(λ)  λ 2 mes(Ω ∩ {f (x) > λ 2 }) (5)

− 1 because too narrow (less than λ 2 ) and infinitely long cusp cannot host eigenvalues less than λ. But for Neumann Laplacian situation is drastically different: essential spectrum appears even if f (x) = e−cx ! Only for f (x) = O(e−xα ) with α > 1 all spectrum is discrete.

Generalizations

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 18 / 39 In fact,

d − 1 N(λ)  λ 2 mes(Ω ∩ {f (x) > λ 2 }) (5)

− 1 because too narrow (less than λ 2 ) and infinitely long cusp cannot host eigenvalues less than λ. But for Neumann Laplacian situation is drastically different: essential spectrum appears even if f (x) = e−cx ! Only for f (x) = O(e−xα ) with α > 1 all spectrum is discrete.

Generalizations

Weyl law understood literally claims that N(λ) = ∞ if mes(Ω) = ∞. But it is not the case for Dirichlet problem: N(λ) < ∞ provided f (x) → 0 as x → +∞.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 18 / 39 But for Neumann Laplacian situation is drastically different: essential spectrum appears even if f (x) = e−cx ! Only for f (x) = O(e−xα ) with α > 1 all spectrum is discrete.

Generalizations

Weyl law understood literally claims that N(λ) = ∞ if mes(Ω) = ∞. But it is not the case for Dirichlet problem: N(λ) < ∞ provided f (x) → 0 as x → +∞. In fact,

d − 1 N(λ)  λ 2 mes(Ω ∩ {f (x) > λ 2 }) (5)

− 1 because too narrow (less than λ 2 ) and infinitely long cusp cannot host eigenvalues less than λ.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 18 / 39 Only for f (x) = O(e−xα ) with α > 1 all spectrum is discrete.

Generalizations

Weyl law understood literally claims that N(λ) = ∞ if mes(Ω) = ∞. But it is not the case for Dirichlet problem: N(λ) < ∞ provided f (x) → 0 as x → +∞. In fact,

d − 1 N(λ)  λ 2 mes(Ω ∩ {f (x) > λ 2 }) (5)

− 1 because too narrow (less than λ 2 ) and infinitely long cusp cannot host eigenvalues less than λ. But for Neumann Laplacian situation is drastically different: essential spectrum appears even if f (x) = e−cx !

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 18 / 39 Generalizations

Weyl law understood literally claims that N(λ) = ∞ if mes(Ω) = ∞. But it is not the case for Dirichlet problem: N(λ) < ∞ provided f (x) → 0 as x → +∞. In fact,

d − 1 N(λ)  λ 2 mes(Ω ∩ {f (x) > λ 2 }) (5)

− 1 because too narrow (less than λ 2 ) and infinitely long cusp cannot host eigenvalues less than λ. But for Neumann Laplacian situation is drastically different: essential spectrum appears even if f (x) = e−cx ! Only for f (x) = O(e−xα ) with α > 1 all spectrum is discrete.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 18 / 39 f.e. Schr¨odingieroperators defined on the base of the cusp but may be with a potential which is a function which values are operators in the 2 auxiliary space H, which is L (Ω) for Dirichlet Laplacian in the domain with a thick cusp, and just C for a Neumann Laplacian in the domain with an ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variables 0 0 x 7→ x /f (x1) then operator −∆ becomes (roughly) such Schr¨odinger 2 −2 operator−∂1 − f (x1) ∆G and separation of variables leads to 2 −2 −∂1 + f (x1) µm where µm are eigenvalues of the Laplacian in G. It is fine for Dirichlet Laplacian (µm > 0) but for Neumann Laplacian we get a problem as µ1 = 0. However more educated change of variables (taking in account the 1 2 measure) leads to an extra potential 4 (∂1 log f ) which saves the day!

Generalizations

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 19 / 39 which is L 2(Ω) for Dirichlet Laplacian in the domain with a thick cusp, and just C for a Neumann Laplacian in the domain with an ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variables 0 0 x 7→ x /f (x1) then operator −∆ becomes (roughly) such Schr¨odinger 2 −2 operator−∂1 − f (x1) ∆G and separation of variables leads to 2 −2 −∂1 + f (x1) µm where µm are eigenvalues of the Laplacian in G. It is fine for Dirichlet Laplacian (µm > 0) but for Neumann Laplacian we get a problem as µ1 = 0. However more educated change of variables (taking in account the 1 2 measure) leads to an extra potential 4 (∂1 log f ) which saves the day!

Generalizations

Remark Even if Weyl’s law understood literally fails in the examples above, the answer is often given in the terms of Weyl’s law for some other operators, f.e. Schr¨odingieroperators defined on the base of the cusp but may be with a potential which is a function which values are operators in the auxiliary space H,

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 19 / 39 and just C for a Neumann Laplacian in the domain with an ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variables 0 0 x 7→ x /f (x1) then operator −∆ becomes (roughly) such Schr¨odinger 2 −2 operator−∂1 − f (x1) ∆G and separation of variables leads to 2 −2 −∂1 + f (x1) µm where µm are eigenvalues of the Laplacian in G. It is fine for Dirichlet Laplacian (µm > 0) but for Neumann Laplacian we get a problem as µ1 = 0. However more educated change of variables (taking in account the 1 2 measure) leads to an extra potential 4 (∂1 log f ) which saves the day!

Generalizations

Remark Even if Weyl’s law understood literally fails in the examples above, the answer is often given in the terms of Weyl’s law for some other operators, f.e. Schr¨odingieroperators defined on the base of the cusp but may be with a potential which is a function which values are operators in the 2 auxiliary space H, which is L (Ω) for Dirichlet Laplacian in the domain with a thick cusp,

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 19 / 39 but for Neumann Laplacian we get a problem as µ1 = 0. However more educated change of variables (taking in account the 1 2 measure) leads to an extra potential 4 (∂1 log f ) which saves the day!

Generalizations

Remark Even if Weyl’s law understood literally fails in the examples above, the answer is often given in the terms of Weyl’s law for some other operators, f.e. Schr¨odingieroperators defined on the base of the cusp but may be with a potential which is a function which values are operators in the 2 auxiliary space H, which is L (Ω) for Dirichlet Laplacian in the domain with a thick cusp, and just C for a Neumann Laplacian in the domain with an ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variables 0 0 x 7→ x /f (x1) then operator −∆ becomes (roughly) such Schr¨odinger 2 −2 operator−∂1 − f (x1) ∆G and separation of variables leads to 2 −2 −∂1 + f (x1) µm where µm are eigenvalues of the Laplacian in G. It is fine for Dirichlet Laplacian (µm > 0)

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 19 / 39 However more educated change of variables (taking in account the 1 2 measure) leads to an extra potential 4 (∂1 log f ) which saves the day!

Generalizations

Remark Even if Weyl’s law understood literally fails in the examples above, the answer is often given in the terms of Weyl’s law for some other operators, f.e. Schr¨odingieroperators defined on the base of the cusp but may be with a potential which is a function which values are operators in the 2 auxiliary space H, which is L (Ω) for Dirichlet Laplacian in the domain with a thick cusp, and just C for a Neumann Laplacian in the domain with an ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variables 0 0 x 7→ x /f (x1) then operator −∆ becomes (roughly) such Schr¨odinger 2 −2 operator−∂1 − f (x1) ∆G and separation of variables leads to 2 −2 −∂1 + f (x1) µm where µm are eigenvalues of the Laplacian in G. It is fine for Dirichlet Laplacian (µm > 0) but for Neumann Laplacian we get a problem as µ1 = 0.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 19 / 39 Generalizations

Remark Even if Weyl’s law understood literally fails in the examples above, the answer is often given in the terms of Weyl’s law for some other operators, f.e. Schr¨odingieroperators defined on the base of the cusp but may be with a potential which is a function which values are operators in the 2 auxiliary space H, which is L (Ω) for Dirichlet Laplacian in the domain with a thick cusp, and just C for a Neumann Laplacian in the domain with an ultra-thin cusp.

Indeed, if we consider cusp as on Figure 5 and make change of variables 0 0 x 7→ x /f (x1) then operator −∆ becomes (roughly) such Schr¨odinger 2 −2 operator−∂1 − f (x1) ∆G and separation of variables leads to 2 −2 −∂1 + f (x1) µm where µm are eigenvalues of the Laplacian in G. It is fine for Dirichlet Laplacian (µm > 0) but for Neumann Laplacian we get a problem as µ1 = 0. However more educated change of variables (taking in account the 1 2 measure) leads to an extra potential 4 (∂1 log f ) which saves the day!

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 19 / 39 Generalizations Variational methods

Variational methods introduced by Weyl were developed by many authors. I want to mention only Mikhail Birman (and his school), Elliott Lieb (and his school) and Barry Simon (and his school).

Mikhail Birman

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 20 / 39 In particular, for semiclassical asymptotics for operator −h2∆ + V (x) as h → +0 we expect Z d − −d 2 Nh ∼ (2πh) ωd V− dx (7)

Generalizations Generalized Weyl law

Z d − −d 2 N ≈ (2π) ωd V− dx (6)

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 21 / 39 Generalizations Generalized Weyl law

Z d − −d 2 N ≈ (2π) ωd V− 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) as h → +0 we expect Z d − −d 2 Nh ∼ (2πh) ωd V− dx (7)

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 21 / 39 Grisha Rozenblum, [Roz1] (1972), proved actually much more general estimate. Other proofs (based on different ideas) were given by Michael Cwikel (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) with semiclassical constant in (6)!

Generalizations Estimates

The famous Cwikel-Lieb-Rozenblum estimate for number of negative d eigenvalues of Schrdingier¨ −∆ + V (x) operator in R (or in domain with Dirichlet boundary conditions:

Z d − 2 N ≤ C V− dx (8)

with C = Cd depending only on d ≥ 3.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 22 / 39 Other proofs (based on different ideas) were given by Michael Cwikel (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) with semiclassical constant in (6)!

Generalizations Estimates

The famous Cwikel-Lieb-Rozenblum estimate for number of negative d eigenvalues of Schrdingier¨ −∆ + V (x) operator in R (or in domain with Dirichlet boundary conditions:

Z d − 2 N ≤ C V− dx (8)

with C = Cd depending only on d ≥ 3. Grisha Rozenblum, [Roz1] (1972), proved actually much more general estimate.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 22 / 39 Bad news: Cd in CLR does not coincide (and cannot coincide) with semiclassical constant in (6)!

Generalizations Estimates

The famous Cwikel-Lieb-Rozenblum estimate for number of negative d eigenvalues of Schrdingier¨ −∆ + V (x) operator in R (or in domain with Dirichlet boundary conditions:

Z d − 2 N ≤ C V− dx (8)

with C = Cd depending only on d ≥ 3. Grisha Rozenblum, [Roz1] (1972), proved actually much more general estimate. Other proofs (based on different ideas) were given by Michael Cwikel (1977), Elliott Lieb (1980), Peter Li and Shing-Tung Yau (1983), Joseph Conlon (1985).

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

The famous Cwikel-Lieb-Rozenblum estimate for number of negative d eigenvalues of Schrdingier¨ −∆ + V (x) operator in R (or in domain with Dirichlet boundary conditions:

Z d − 2 N ≤ C V− dx (8)

with C = Cd depending only on d ≥ 3. Grisha Rozenblum, [Roz1] (1972), proved actually much more general estimate. Other proofs (based on different ideas) were given by Michael Cwikel (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) with semiclassical constant in (6)!

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 22 / 39 CLR implies Lieb–Thirring inequality

Z d X 0 2 +1 |λj | ≤ C V− dx (9) λj <0

0 albeit constant could be better than trivial Cd = 2Cd /(d + 2).

Generalizations

CLR however could be used in conjugation with variational and Tauberian methods to prove asymptotics, including those with sharp remainder estimates. CLR is used to prove that certain “singular” zones provide small contribution to eigenvalue counting function.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 23 / 39 Generalizations

CLR however could be used in conjugation with variational and Tauberian methods to prove asymptotics, including those with sharp remainder estimates. CLR is used to prove that certain “singular” zones provide small contribution to eigenvalue counting function. CLR implies Lieb–Thirring inequality

Z d X 0 2 +1 |λj | ≤ C V− dx (9) λj <0

0 albeit constant could be better than trivial Cd = 2Cd /(d + 2).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 23 / 39 If ψ = 1, we get N(λ) in the right-hand expression, but generally we want to assemble 1 from differently scaled ψι.

Variants Local Weyl Law

Local Weyl Law:

Z Z d −d 2 e(x, x, λ)ψ(x) dx = Tr E(λ)ψ ≈ (2π) ωd V− ψ(x) dx (10)

where e(x, y, λ) is the Schwartz’ kernel of the spectral projector E(λ) of ∞ operator H = −∆ + V and ψ(x) is a C0 cut-off function.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 24 / 39 Variants Local Weyl Law

Local Weyl Law:

Z Z d −d 2 e(x, x, λ)ψ(x) dx = Tr E(λ)ψ ≈ (2π) ωd V− ψ(x) dx (10)

where e(x, y, λ) is the Schwartz’ kernel of the spectral projector E(λ) of ∞ operator H = −∆ + V and ψ(x) is a C0 cut-off function. If ψ = 1, we get N(λ) in the right-hand expression, but generally we want to assemble 1 from differently scaled ψι.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 24 / 39 If negative spectrum of H is purely point, then X e(x, y, 0) = φj (x)φj (y) (12)

λj <0

where φj are orthonormal eigenfunctions, corresponding to eigenvalues λj .

Variants Pointwise Weyl Law

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

d −d 2 e(x, x, 0) ≈ (2π) ωd V− (x) (11) where e(x, y, λ) is the Schwartz kernel of the spectral projector of H = −∆ + V (x).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 25 / 39 Variants Pointwise Weyl Law

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

d −d 2 e(x, x, 0) ≈ (2π) ωd V− (x) (11) where e(x, y, λ) is the Schwartz kernel of the spectral projector of H = −∆ + V (x). If negative spectrum of H is purely point, then X e(x, y, 0) = φj (x)φj (y) (12)

λj <0

where φj are orthonormal eigenfunctions, corresponding to eigenvalues λj .

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 25 / 39 2 3 This operator acts on the space ∧1≤j≤N L (R ) of totally antisymmetric functions Ψ(x1,..., xN ) because electrons are Fermions and we ignore spin.

Variants Application: Thomas-Fermi Theory

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

X X Zm X 1 HN = −∆xj − + (13) |xj − ym| |xj − xk | 1≤j≤N 1≤m≤M, 1≤j

1 where Planck constant ~ = 1, electron mass = 2 , electron charge = −1, ym is a location of m-th nuclei and Zm its charge, M is fixed, but Zm  N → ∞.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 26 / 39 Variants Application: Thomas-Fermi Theory

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

X X Zm X 1 HN = −∆xj − + (13) |xj − ym| |xj − xk | 1≤j≤N 1≤m≤M, 1≤j

1 where Planck constant ~ = 1, electron mass = 2 , electron charge = −1, ym is a location of m-th nuclei and Zm its charge, M is fixed, but Zm  N → ∞. 2 3 This operator acts on the space ∧1≤j≤N L (R ) of totally antisymmetric functions Ψ(x1,..., xN ) because electrons are Fermions and we ignore spin.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 26 / 39 P 2 Then the local electron density would be ρΨ = 1≤j≤N |φj (x)| and according to pointwise Weyl law

1 3 ρ (x) ≈ (W + ν) 2 (14) Ψ 6π2 +

where ν = λN . −1 This density would generate potential −|x| ∗ ρΨ and we would have −1 W ≈ V − |x| ∗ ρΨ.

Variants

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 27 / 39 and according to pointwise Weyl law

1 3 ρ (x) ≈ (W + ν) 2 (14) Ψ 6π2 +

where ν = λN . −1 This density would generate potential −|x| ∗ ρΨ and we would have −1 W ≈ V − |x| ∗ ρΨ.

Variants

If electrons were not interacting between themselves but the field potential was −W (x) then they would occupy lowest eigenvalues and ground state wave functions would be (anti-symmetrized) φ1(x1)φ2(x2) . . . φN (xN ) where φj and λj are eigenfunctions and eigenvalues of H = −∆ − W (x). P 2 Then the local electron density would be ρΨ = 1≤j≤N |φj (x)|

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 27 / 39 −1 This density would generate potential −|x| ∗ ρΨ and we would have −1 W ≈ V − |x| ∗ ρΨ.

Variants

If electrons were not interacting between themselves but the field potential was −W (x) then they would occupy lowest eigenvalues and ground state wave functions would be (anti-symmetrized) φ1(x1)φ2(x2) . . . φN (xN ) where φj and λj are eigenfunctions and eigenvalues of H = −∆ − W (x). P 2 Then the local electron density would be ρΨ = 1≤j≤N |φj (x)| and according to pointwise Weyl law

1 3 ρ (x) ≈ (W + ν) 2 (14) Ψ 6π2 + where ν = λN .

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 27 / 39 Variants

If electrons were not interacting between themselves but the field potential was −W (x) then they would occupy lowest eigenvalues and ground state wave functions would be (anti-symmetrized) φ1(x1)φ2(x2) . . . φN (xN ) where φj and λj are eigenfunctions and eigenvalues of H = −∆ − W (x). P 2 Then the local electron density would be ρΨ = 1≤j≤N |φj (x)| and according to pointwise Weyl law

1 3 ρ (x) ≈ (W + ν) 2 (14) Ψ 6π2 + where ν = λN . −1 This density would generate potential −|x| ∗ ρΨ and we would have −1 W ≈ V − |x| ∗ ρΨ.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 27 / 39 Thomas-Fermi theory has been rigorously justified (with pretty good error estimates).

Variants

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

V − W TF = |x|−1 ∗ ρTF, (15) 1 3 ρTF = (W TF + ν) 2 , (16) 6π2 + Z ρTF dx = N (17) where ν ≤ 0 is called chemical potential and in fact approximates λN .

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 28 / 39 Variants

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

V − W TF = |x|−1 ∗ ρTF, (15) 1 3 ρTF = (W TF + ν) 2 , (16) 6π2 + Z 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 error estimates).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 28 / 39 and justified Scott correction term ∼ N2 and Dirac and Schwinger 5 5 −δ correction terms ∼ N 3 (so the error is O(N 3 ) with some δ > 0). More details (including models with magnetic field – either external or self-generated) are in my talk

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

Variants

In fact the ground state energy is given by

2 5 Z (6π ) 3  TF 5 TF E = ρ 3 − V ρ dx N 15π2 1 − ρTF(x)ρTF(y)|x − y|−1 dxdy + O(N2) (18) 2 x

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 29 / 39 More details (including models with magnetic field – either external or self-generated) are in my talk

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

Variants

In fact the ground state energy is given by

2 5 Z (6π ) 3  TF 5 TF E = ρ 3 − V ρ dx N 15π2 1 − ρTF(x)ρTF(y)|x − y|−1 dxdy + O(N2) (18) 2 x and justified Scott correction term ∼ N2 and Dirac and Schwinger 5 5 −δ correction terms ∼ N 3 (so the error is O(N 3 ) with some δ > 0).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 29 / 39 Variants

In fact the ground state energy is given by

2 5 Z (6π ) 3  TF 5 TF E = ρ 3 − V ρ dx N 15π2 1 − ρTF(x)ρTF(y)|x − y|−1 dxdy + O(N2) (18) 2 x and justified Scott correction term ∼ N2 and Dirac and Schwinger 5 5 −δ correction terms ∼ N 3 (so the error is O(N 3 ) with some δ > 0). More details (including models with magnetic field – either external or self-generated) are in my talk

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 29 / 39 For example, if domain has certain symmetries we can consider subspaces of functions having “similar” symmetries. This could be crucial: for example, Neumann Laplacian in the domain with a cusp (which is not ultra-thin) has an essential spectrum. However, if this domain is symmetric with respect to the plane passing through the middle of the cusp, and we consider only functions which are odd, the essential spectrum disappears and operator has discrete spectrum. Another example: in the simplest problems arising in electromagnetic oscillations and in hydrodynamics we need to consider −∆ on the space of solenoidal fields: ∇ · u = 0. Crystal optics provides more complicated examples.

Variants Going to subspaces

In many problems of mathematical physics the domain of operator is dense 2 m not in L (Ω, R ) but in some its subspace.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 30 / 39 Another example: in the simplest problems arising in electromagnetic oscillations and in hydrodynamics we need to consider −∆ on the space of solenoidal fields: ∇ · u = 0. Crystal optics provides more complicated examples.

Variants Going to subspaces

In many problems of mathematical physics the domain of operator is dense 2 m not in L (Ω, R ) but in some its subspace. For example, if domain has certain symmetries we can consider subspaces of functions having “similar” symmetries. This could be crucial: for example, Neumann Laplacian in the domain with a cusp (which is not ultra-thin) has an essential spectrum. However, if this domain is symmetric with respect to the plane passing through the middle of the cusp, and we consider only functions which are odd, the essential spectrum disappears and operator has discrete spectrum.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 30 / 39 Crystal optics provides more complicated examples.

Variants Going to subspaces

In many problems of mathematical physics the domain of operator is dense 2 m not in L (Ω, R ) but in some its subspace. For example, if domain has certain symmetries we can consider subspaces of functions having “similar” symmetries. This could be crucial: for example, Neumann Laplacian in the domain with a cusp (which is not ultra-thin) has an essential spectrum. However, if this domain is symmetric with respect to the plane passing through the middle of the cusp, and we consider only functions which are odd, the essential spectrum disappears and operator has discrete spectrum. Another example: in the simplest problems arising in electromagnetic oscillations and in hydrodynamics we need to consider −∆ on the space of solenoidal fields: ∇ · u = 0.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 30 / 39 Variants Going to subspaces

In many problems of mathematical physics the domain of operator is dense 2 m not in L (Ω, R ) but in some its subspace. For example, if domain has certain symmetries we can consider subspaces of functions having “similar” symmetries. This could be crucial: for example, Neumann Laplacian in the domain with a cusp (which is not ultra-thin) has an essential spectrum. However, if this domain is symmetric with respect to the plane passing through the middle of the cusp, and we consider only functions which are odd, the essential spectrum disappears and operator has discrete spectrum. Another example: in the simplest problems arising in electromagnetic oscillations and in hydrodynamics we need to consider −∆ on the space of solenoidal fields: ∇ · u = 0. Crystal optics provides more complicated examples.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 30 / 39 Assume that V is periodic or quasi periodic (whatever it means) and d domain is R . Let Ω be a box or a ball centred at origin, consider stretched domain LΩ and consider either 1 Z N(λ) = lim e(x, x, λ) dx (19) L→∞ mes(LΩ) LΩ or 1 N(λ) = lim NLΩ(λ) (20) L→∞ mes(LΩ)

d where in the former case e(x, y, λ) is defined for R but integrated over R LΩ 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 quantity replacing N(λ) we can study?

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 31 / 39 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 quantity replacing N(λ) we can study? Assume that V is periodic or quasi periodic (whatever it means) and d domain is R . Let Ω be a box or a ball centred at origin, consider stretched domain LΩ and consider either 1 Z N(λ) = lim e(x, x, λ) dx (19) L→∞ mes(LΩ) LΩ or 1 N(λ) = lim NLΩ(λ) (20) L→∞ mes(LΩ)

d where in the former case e(x, y, λ) is defined for R but integrated over R LΩ and in the letter NLΩ(λ) = LΩ eLΩ(x, x, λ) dx is calculated for LΩ.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 31 / 39 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 quantity replacing N(λ) we can study? Assume that V is periodic or quasi periodic (whatever it means) and d domain is R . Let Ω be a box or a ball centred at origin, consider stretched domain LΩ and consider either 1 Z N(λ) = lim e(x, x, λ) dx (19) L→∞ mes(LΩ) LΩ or 1 N(λ) = lim NLΩ(λ) (20) L→∞ mes(LΩ)

d where in the former case e(x, y, λ) is defined for R but integrated over R LΩ and in the letter NLΩ(λ) = LΩ eLΩ(x, x, λ) dx is calculated for LΩ. Usually these definitions are equivalent.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 31 / 39 Variants Integrated density of states

There are problems when spectrum is essential. What global quantity replacing N(λ) we can study? Assume that V is periodic or quasi periodic (whatever it means) and d domain is R . Let Ω be a box or a ball centred at origin, consider stretched domain LΩ and consider either 1 Z N(λ) = lim e(x, x, λ) dx (19) L→∞ mes(LΩ) LΩ or 1 N(λ) = lim NLΩ(λ) (20) L→∞ mes(LΩ)

d where in the former case e(x, y, λ) is defined for R but integrated over R LΩ 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 H and H0 are close at infinity. Then one can study Krein-Birman spectral shift function Z
eH (x, x, λ) − eH0 (x, x, λ) dx (21)

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

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 32 / 39 Variants ReferenceI

W. Arendt W., R. Nittka R., W. Peter W. and F. Steiner. Weyls Law: Spectral Properties of the Laplacian in Mathematics and Physics, pp. 1–71, in Mathematical Analysis of Evolution, Information, and Complexity, by W. Arendt and W.P. Schleich, Wiley-VCH, 2009. V. G. Avakumovicˇ. Uber¨ die eigenfunktionen auf geschlossen riemannschen mannigfaltigkeiten. Math. Z., 65:324–344 (1956). T. Carleman. Propri´etesasymptotiques des fonctions fondamentales des membranes vibrantes. In C. R. 8-`emeCongr. Math. Scand., Stockholm, 1934, pages 34–44, Lund (1935). T. Carleman. Uber¨ die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen. Ber. Sachs. Acad. Wiss. Leipzig, 88:119–132 (1936).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 33 / 39 Variants ReferenceII

R. Courant.Uber¨ die Eigenwerte bei den Differentialgleichungen der mathematischen Physik. Mat. Z., 7:1–57 (1920). J. J. Duistermaat and V. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math., 29(1):37–79 (1975). J. J. Duistermaat and V. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math., 29(1):37–79 (1975). L. Hormander¨ . The spectral function of an elliptic operator. Acta Math., 121:193–218 (1968).

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 34 / 39 Variants ReferenceIII

L. Hormander¨ . On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators. In Yeshiva Univ. 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 the Laplace-Beltrami operator on manifold with boundary. Funct. Anal. Appl., 14(2):98–106 (1980). V. Ivrii. Accurate spectral asymptotics for elliptic operators that act in vector bundles. Funct. Anal. Appl., 16(2):101–108 (1982). V. Ivrii. Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, SMM, 1998, xv+731.

Victor Ivrii (Math., Toronto) 100 years of Weyl’s law April 22, 2017 35 / 39 Variants ReferenceIV

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