Sum rules and semiclassical limits for the spectra of some elliptic PDEs and pseudodifferential operators

Transcription

1 Sum rules and semiclassical limits for the spectra of some elliptic PDEs and pseudodifferential operators Evans Harrell Georgia Tech Institut É. Cartan Univ. H. Poincaré Nancy 1 16 février, 2010 Copyright 2010 by Evans M. Harrell II.

2 Laplace, Beltrami, and Schrödinger. H = T + V(x)

3 Laplace, Beltrami, and Schrödinger. - H = T + V(x) H φ k = λ k φ k

4 What does semiclassical mean?

5 A Schrödinger operator with correct physical numbers. H = 2 2m + V (x) k

6 A Schrödinger operator with correct physical numbers. H = 2 2m + V (x) k = J-s m = kg The coefficient ε of the Laplacian is not large.

7 Various semiclassical limits Let or equivalently ε tend to 0. Put a large parameter in front of V. Consider high energies (k large for λ k ). Consider many particles, like

8 Semiclassical limits 1. λ k 2. H = εt + V(x), (ε small)

9 Mathematical motivation: Not just any sequence of positive numbers can be the spectrum of the Dirichlet Laplacian on a bounded domain. Similarly for plausible spectra of Schrödinger Operators.

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11 There s something Strang about this column!

12 Should you believe this man? For the 128-gon,

13 Should you believe this man? For the 128-gon,

14 Should you believe this man? For the 128-gon,

15 But according to the Ashbaugh-Benguria Theorem, λ 2 / λ ! (In 2 D)

16 Aha!

17 Universal constraints on the H. Weyl, 1910, Laplace, λ n ~ n 2/d spectrum

18 Universal constraints on the spectrum H. Weyl, 1910, Laplace, λ n ~ n 2/d W. Kuhn, F. Reiche, W. Thomas, W. Heisenberg, 1925, sum rules for atomic energies.

19 Universal constraints on the spectrum H. Weyl, 1910, Laplace, λ n ~ n 2/d W. Kuhn, F. Reiche, W. Thomas, W. Heisenberg, 1925, sum rules for atomic energies. L. Payne, G. Pólya, H. Weinberger, 1956: The gap is controlled by the average of the smaller eigenvalues:

20 Universal constraints on the Ashbaugh-Benguria 1991, isoperimetric conjecture of PPW proved. spectrum H. Yang 1991, unpublished, formulae like PPW, respecting Weyl asymptotics for the first time. Harrell 1993-present, commutator approach; with Michel, Stubbe, El Soufi and Ilias, Hermi, Yildirim. Ashbaugh-Hermi, Levitin-Parnovsky, Cheng- Yang, Cheng-Chen, some others.

21 Universal constraints on the spectrum with phase-space volume. Lieb -Thirring, 1977, for Schrödinger ɛ d/2 λ j (ɛ) ρ L ρ,d λ j (ɛ)<0 Li - Yau, 1983 (Berezin 1973), for Laplace R d (V (x)) ρ+d/2 dx k j=1 λ j d d + 2 4π 2 k 1+2/d (C d Ω ) 2/d

22 Stubbe s proof of sharp Lieb- Thirring for ρ 2 (JEMS, in press)

23 Stubbe s proof of sharp Lieb- Thirring for ρ 2 (JEMS, in press) 1. A trace formula ( sum rule ) of Harrell-Stubbe 97, for H = - ε Δ + V: R ρ (z) := (z λ k ) ρ +; R ρ (z) ɛ 2ρ d (z λk ) ρ 1 + φ k 2 = explicit expr 0. ( )

24 1. A trace formula ( sum rule ) of Harrell-Stubbe 97, for H = - ε Δ + V: 2. Stubbe s proof of sharp Lieb- Thirring for ρ 2 (JEMS, in press) R ρ (z) ɛ 2ρ d T k := φ k, φ k R ρ (z) := (z λ k ) ρ +; (z λk ) ρ 1 + φ k 2 = explicit expr 0. ( )

25 1. A trace formula ( sum rule ) of Harrell-Stubbe 97, for H = - ε Δ + V: 2. Stubbe s proof of sharp Lieb- Thirring for ρ 2 (JEMS, in press) R ρ (z) ɛ 2ρ d T k := φ k, φ k R ρ (z) := (z λ k ) ρ +; (z λk ) ρ 1 + φ k 2 = explicit expr 0. ( ) = dλ k dɛ (Feynman-Hellman)

26 Thus or: Lieb-Thirring inequalities and classical Lieb-Thirring is an immediate consequence! Recall: ion: lim ɛ 0 + ɛ d 2 λ j (ɛ)<0 R ρ (z, ɛ) 2ɛ R ρ (z, ɛ), d ɛ ( ) ɛ d 2 Rρ (z, ɛ) 0, ɛ λ j (ɛ) ρ = L ρ,d V( x) ρ+ d 2

27 Some models in nanophysics: 1. Schrödinger operators on curves and surfaces embedded in space. Quantum wires and waveguides. 2. Periodic Schrödinger operators. Electrons in crystals. 3. Quantum graphs. Nanoscale circuits 4. Relativistic Hamiltonians on curved surfaces. Graphene.

28 Are the spectra of these models controlled by sum rules, like those known for Laplace/Schrödinger on domains or all of R d, or are there important differences?

29 Are the spectra of these models controlled by sum rules? If so, can we prove analogues of Lieb-Thirring, Li-Yau, PPW, etc.?

30 Sum Rules 1. Used by Heisenberg in 1925 to explain regularities in atomic energy spectra

31 Sum Rules 1. Observations by Thomas, Reiche, Kuhn of regularities in atomic energy spectra. 2. Heisenberg,1925, Showed TRK purely algebraic, following from noncommutation of operators. 3. Bethe, 1930, other identities.

32 Commutators of operators [H, G] := HG - GH [H, G] φ k = (H - λ k ) G φ k If H=H*, <φ j,[h, G] φ k > = (λ j - λ k ) <φ j,gφ k >

33 Commutators of operators [G, [H, G]] = 2 GHG - G 2 H - HG 2 Etc., etc. Typical consequence: φ j, [G, [H, G]]φ j = k:λ k λ j (λ k λ j ) G kj 2 (Abstract version of Bethe s sum rule)

34 Riesz means The counting function, N(z) := #(λ k z) Integrals of the counting function, known as Riesz means (Safarov, Laptev, Weidl, etc.): Chandrasekharan and Minakshisundaram, 1952

35 1 st and 2 nd commutators (H-S 97) 1 2 λ j J λ j J (z λ j ) 2 [G, [H, G]]φ j, φ j λ j J = λ k J c (z λ j )(z λ k )(λ k λ j ) Gφ j, φ k 2 (z λ j ) [H, G]φ j 2 The only assumptions are that H and G are selfadjoint, and that the eigenfunctions are a complete orthonormal sequence. (If continuous spectrum, need a spectral integral on right.)

36 ( ) ( ) 1 2 λ j J λ j J { } Or even without G=G*: (z λ j ) 2 ( [G, [H, G]]φ j, φ j + [G, [H, G ]]φ j, φ j ) (z λ j ) ( [H, G]φ j, [H, G]φ j + [H, G ]φ j, [H, G ]φ j ) = λ j J λ k / J (z λ j )(z λ k )(λ k λ j ) ( Gφ j, φ k 2 + G φ j, φ k 2),

37 ( ) ( ) 1 2 λ j J λ j J { } Or even without G=G*: (z λ j ) 2 ( [G, [H, G]]φ j, φ j + [G, [H, G ]]φ j, φ j ) (z λ j ) ( [H, G]φ j, [H, G]φ j + [H, G ]φ j, [H, G ]φ j ) = λ j J λ k / J (z λ j )(z λ k )(λ k λ j ) ( Gφ j, φ k 2 + G φ j, φ k 2), When does this side have a sign?

38 What you should remember about trace formulae/sum rules in a short seminar?

39 What you should remember about trace formulae/sum rules in a short seminar? 1. There is an exact identity involving traces including [G, [H, G]] and [H,G]*[H,G]. 2. For the lower part of the spectrum it implies an inequality of the form: (z λ k ) 2 (...) (z λ k ) (...)

40 Universal bounds for Dirichlet Laplacians Yang 1991: { Payne-Pólya-Weinberger, 1956: λ k+1 λ k 4 d Hile-Protter 1980: 1 4 d 1 k k j=1 1 k λ j k λ j =: 4 d λ k j=1 λ k+1 λ j }

41 Universal Bounds with Commutators Hile-Protter vs. sum rule (H-S 97):

42 Dirichlet problem: Trace identities imply differential inequalities Harrell-Hermi JFA 08: Laplacian R 2 (z) 4 d ( ) (z λ k )T k k Consequences universal bound for k >j: λ k 4 + d λ j 2 + d ( k j ) 2/d

43 Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean.

44 D k := Statistics of spectra

45 How to get information about eigenvalues from information on Riesz means?

46 How to get information about eigenvalues from information on Riesz means? Riesz means are related to:

47 How to get information about eigenvalues from information on Riesz means? Riesz means are related to sums of eigenvalues by Legendre transform

48 How to get information about eigenvalues from information on Riesz means? Riesz means are related to sums of eigenvalues by Legendre transform partition functions by Laplace transform

49 Some models in nanophysics: 1. Schrödinger operators on curves and surfaces embedded in space. Quantum wires and waveguides. 2. Periodic Schrödinger operators. Electrons in crystals. 3. Quantum graphs. Nanoscale circuits 4. Relativistic Hamiltonians on curved surfaces. Graphene.

50 In each of the four models there are new features in the trace inequality. 1. Schrödinger operators on curves and surfaces. Explicit curvature terms. 2. Periodic Schrödinger operators. Geometry of the dual lattice. 3. Quantum graphs. Topology 4. Relativistic Hamiltonians. First-order ΨDO rather than second-order.

51 Klein-Gordon operators, a.k.a., generators of Cauchy processes 1. Motivated by graphene: electrons are relativistic, albeit with c/ On infinite R 2, H = Dirac operator or (- Δ + m 2 ) 1/2 (Klein Gordon). 3. When there are edges, imposition of BC not natural from PDE point of view.

52 Eigenvalue inequalities for Klein-Gordon Operators Evans M. Harrell II School of Mathematics, Georgia Institute of Technology, Atlanta, GA U.S.A. Selma Yıldırım Yolcu School of Mathematics, Georgia Institute of Technology, Atlanta, GA U.S.A. Abstract We consider the pseudodifferential operators H m,ω associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian P 2 + m 2 when restricted to a bounded, open domain Ω R d. When the mass m is 0 the operator H 0,Ω coincides with the generator of the Cauchy stochastic process with a killing condition on Ω. (The operator H 0,Ω is sometimes called the fractional Laplacian with power 1 2, cf. [15].) We prove several universal inequalities for the eigenvalues 0 < β 1 < β 2 of H m,ω and their means β k := 1 k k l=1 β l. Among the inequalities proved are: ( )

53 given by ϕ := F ξ ϕ(ξ), and therefor + m2 ϕ := F 1 ξ 2 + m 2 ϕ(ξ). R exp ( t) [ϕ] (x) = p 0 (t, ) ϕ, p 0 (t, x) := c d t (t 2 + x 2 ) d+1 2,!

54 Definition of K-G: Calculate the square root of - Δ + m 2, and afterwards restrict to Ω.

55 Definition of K-G: Calculate the square root of - Δ + m 2, and afterwards restrict to Ω. Not the same as restricting to Ω with DBC and then taking square root by spectral methods!

56 Comparison to the free Laplacian ϕ, H 2 m,ωϕ = H m,ω ϕ 2 = = = = Ω R d R d F 1 ( ξ 2 + m 2 ˆϕ ) 2 χ ΩF 1 ( ξ 2 + m 2 ˆϕ F 1 ( ξ 2 + m 2 ˆϕ R d ϕ( + m 2 )ϕ Ω ϕ( + m 2 )ϕ, ) 2 ) 2 Therefore β k λ k + m 2.

57 Weyl asymptotic for H Ω,m Proposition 3.1 As β, N(β) Ω (4π) d/2 Γ(1 + d/2) βd. Equivalently, as k, β k 4π ( Γ(1 + d/2)k Ω ) 1/d.

58 Among the inequalities proved are: β k cst. ( ) k 1/d Ω for an explicit, optimal semiclassical constant depending only on the dimension d. For any dimension d 2 and any k, Furthermore, when d 2 and k 2j, β k β j β k+1 d + 1 d 1 β k. d 2 1/d (d 1) ( ) 1 k d. j Finally, we present some analogous estimates allowing for an operator including an external potential energy field, i.e, H m,ω + V (x), for V (x) in certain function classes.

59 Calculate first and second commutators: F F [H m,ω, x α ] ϕ = (H m,ω x α x α H m,ω )ϕ = χ Ω F 1 ξ 2 + m 2 F[x α ϕ] χ Ω x α F 1 [ = χ Ω F 1 [ ξ 2 + m 2 ˆϕ ξ α = iχ Ω F 1 ξ α ξ 2 + m ˆϕ. 2 ξ α ( ξ 2 + m 2 ˆϕ) ξ 2 + m 2 ˆϕ] ] Similarly, [x α, [H m,ω, x α ]]ϕ = χ Ω F 1 1 ξ 2 + m 2 2 ξ α ( ξ 2 + m 2 ) 3/2 ˆϕ.

60 Summing over coordinates: (d 1) n j=1 provided z [β n, β n+1 ]. Because (z β j ) 2 u j, Hm,Ωu 1 j 2 n j=1 (z β j ) 0, or, equivalently, (d 1)β 1 n z 2 2dz + (d + 1)β n 0.

61 β n+1 d + 1 (d 1)β 1 n d + 1 d 1 β n. In particular, β 2 β 1 d + 1 d 1,

62 Corollary 2.4 For k > 2j, Eq. (2.24) implies β k β j d 2 1/d (d 1) ( ) 1 k d. j ( )

63 Quantum graphs ( ) (With S. Demirel, Stuttgart.) For which graphs is: R σ (z) L cl σ,1 (V(x) z) σ+1/2 dx? Γ (Concentrate on σ=2.)

64 ON SEMICLASSICAL AND UNIVERSAL INEQUALITIES FOR EIGENVALUES OF QUANTUM GRAPHS SEMRA DEMIREL AND EVANS M. HARRELL II Abstract. We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-Pólya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false.

65 Quantum graphs 1. A graph (in the sense of network) with a 1-D Schrödinger operator on the edges: connected by Kirchhoff conditions at vertices. Sum of outgoing derivatives vanishes.

66 Quantum graphs Is this one-dimensional or not? Does the topology matter?

67 1. Trees. Quantum graphs are onedimensional for:

68 1. Trees. Quantum graphs are onedimensional for: 2. Scottish tartans (infinite rectangular graphs):

69 1. Trees. 2. Infinite rectangular graphs. Quantum graphs are onedimensional for: 3. Bathroom tiles, a.k.a. honeycombs, etc.:

70 Quantum graphs: 1. But not balloons! (A.k.a. tadpoles, or...)

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72 Quantum graphs 1. But not balloons! (A.k.a. tadpoles, or...) ρ = 3/2: ratio is 3/11 vs. L cl = 3/16. ρ = 2: ratio is messy expression vs. L cl = 8/(15 ) =

73 Quantum graphs For which finite graphs is: λ k 4 + d λ j 2 + d ( k j ) 2/d? e.g., is λ 2 /λ 1 5?

74 1. Trees. Quantum graphs

75 Quantum graphs 1. Trees. 2. Rectangular graphs/bathroom tiles with external edges:

76 But not balloons! Quantum graphs

77 Quantum graphs Fancy balloons can have arbitrarily large λ 2 /λ 1.

78 Why?

79 Why? If we can establish the analogue of the trace inequality, R ρ (z) α 2ρ d (z λk ) ρ 1 + φ k 2 0, ( ) then all the rest of the inequalities follow (LT, PPW, ratios, statistics, etc.), sometimes with modifications.

80 Why? If we can establish the analogue of the trace inequality, R ρ (z) α 2ρ d (z λk ) ρ 1 + φ k 2 0, ( ) then all the rest of the inequalities follow (LT, PPW, ratios, statistics, etc.), sometimes with modifications. Calculate commutators with a good G.

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87 THE END