Sum rules and semiclassical limits for quantum Hamiltonians on surfaces, periodic structures, and graphs

Transcription

1 Sum rules and semiclassical limits for quantum Hamiltonians on surfaces, periodic structures, and graphs Evans Harrell Georgia Tech Centre Bernoulli École Polytechnique Fédérale de Lausanne 25 février, 2010 Copyright 2010 by Evans M. Harrell II.

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3 Discrete spectra of Laplace and Schrödinger operators. H = 2 2m + V (x) k H φ k = λ k φ k

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

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

6 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.

7 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:

8 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 spectrum H. Yang 1991, unpublished, formulae like PPW, respecting Weyl asymptotics for the first time.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.

9 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

10 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

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

12 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. ( )

13 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. ( )

14 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)

15 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

16 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.

17 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?

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

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

20 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.

21 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 >

22 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)

23 1 st and 2 nd commutators 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 Harrell-Stubbe TAMS 1997 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.)

24 ( ) ( ) 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),

25 ( ) ( ) 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?

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

27 Take-away messages #1 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 ) (...)

28 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 }

29 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

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

32 Statistics of spectra ( ) 1 = 4 d φ j, φ l λ l λ j l:λ l λ j 2 Harrell-Stubbe TAMS 1997

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

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

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

36 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

37 Take-away messages #2 1. A good choice of G for the Laplacian is a coordinate function, because a) [H,G] = - 2 / x k, and b) [G, [H, G]] = 2 2. For Schrödinger, sum rules conect eigenvalues with the kinetic energy. 3. Spectral information can be extracted from Riesz means with classical transforms.

38 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.

39 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.

40 On a (hyper) surface, what object is most like the Laplacian? (Δ = the good old flat scalar Laplacian of Laplace)

41 Answer #1 (Beltrami s answer): Consider only tangential variations.

42 Difficulty: The Laplace-Beltrami operator is an intrinsic object, and as such is unaware that the surface is immersed!

43 Answer #2 The nanophysicists answer - Δ LB + q, Where the effective potential q responds to how the surface is immersed in space.

44 The result: - Δ LB + q, q(x) = 1 4 ( d ) 2 κ j 1 2 j=1 [ ] d j=1 κ 2 j

45 Heisenberg's Answer (if he had thought about it) q(x) = 1 4 [ ] ( d j=1 κ j ) 2

46 Quadratic sum rule with curvature A good choice of G = x k, a Euclidean coordinate from R d restricted to the submanifold.

47 Quadratic sum rule with curvature A good choice of G = x k, a Euclidean coordinate from R d restricted to the submanifold. There are messy terms, but when you sum the trace identity over k = 1...d, magical cancellations occur.

48 Quadratic sum rule with curvature A good choice of G = x k, a Euclidean coordinate from R d restricted to the submanifold. There are messy terms, but when you sum the trace identity over k = 1...d, magical cancellations occur. Since there are second derivatives of x k, there is a curvature contribution that doesn t go away.

49 Quadratic sum rule with curvature where now T k := R 2 (z) 4 d (z λk ) + T k, ( d ( φ k, + ( j κ ) j) 2 φ k 4 [ ]

50 Quadratic sum rule with curvature Sum rules imply universal bounds on eigenvalue gaps for Schrödinger operators on closed submanifolds in terms of the lower spectrum. Let δ := sup M ( P κ j ) 2 : 4 V (x).

51 Quadratic sum rule with curvature Sum rules imply universal bounds on eigenvalue gaps for Schrödinger operators on closed submanifolds in terms of the lower spectrum. Let δ := sup M ( P κ j ) 2 : 4 V (x).

52 Quadratic sum rule with curvature Sum rules imply universal bounds on eigenvalue gaps for Schrödinger operators on closed submanifolds in terms of the lower spectrum. Let δ := sup M ( P κ j ) 2 : Simplest case is λ 2 λ 1 4 d (λ 1 + δ) 4 V (x). ( )

53 An interesting model H g := + g ( j κ j ) 2

54 An explicit calculation shows that the bound is sharp for the non-zero eigenvalue gaps of the sphere, for which all the eigenvalues are known and elementary [20]: For simplicity, assume that d = 2, g = 1 4, and that M is the sphere of radius 1 embedded in R 3. Then h = 2, σ = 1, and: λ 1 = 1; λ 2 = λ 3 = λ 4 = 3;... ; λ (m 1) 2 +1 = = λ m 2 = m2 m + 1. For n = m 2, the calculation shows that λ n = n+1 2, and λ2 n = n2 +n+1 3. Hence D n = n, and b) informs us that 2λ m 2 m = m 2 m + 1 λ m 2 = m 2 m + 1 λ m 2 +1 = m 2 + m + 1 2λ m 2 + m = m 2 + m + 1, and thus λ m 2 equals the lower bound 2λ m 2 m and λ m 2 +1 equals the upper bound 2λ m 2 + m. q.e.d.

55 What about Lieb-Thirring?

56 What about Lieb-Thirring? Can establish establish a quadratic Yang-type inequality, either by commuting with coordinate functions, or by commuting with unitary operators G = exp(i z.x) (use modified trace identity). Because of the curvature terms, the natural L-T inequality is not in reference to energy 0. Harrell-Stubbe TAMS to appear

57 What about Lieb-Thirring? Similar results for periodic Schrödinger. There is a shift reflecting the periodicity lattice, analogous to the mean curvatures. Geometrically, the periodicities are connected with the curvature necessary to embed a torus in Euclidean space. Harrell-Stubbe TAMS to appear

58 What about Lieb-Thirring? For semiclassics we need a partial differential inequality: ɛ ( ɛ d/2 R 2 ((z, ɛ) ) gd ) 2 ɛd/2 R 1 (z, ɛ) R 2 / z = 2R 1, Harrell-Stubbe TAMS to appear

59 ) What about Lieb-Thirring? see that (??) can be U(z, ɛ) := ɛ d/2 R 2 (z, ɛ) U ɛ gd 4 ξ := ɛ 4 gd z, U z, U ξ 0, (gd reflects the curvature (= sup h 2 ) or, resp. periodicity.) ( Harrell-Stubbe TAMS to appear

60 What about Lieb-Thirring? For all ɛ > 0 the mapping ɛ ɛ d 2 Rσ (z ɛgd 4 ) = ɛ d ɛgd 2 (z 4 λ j) σ + is nonincreasing, and therefore for all z R and all ɛ > 0 the following sharp Lieb-Thirring inequality holds: ( R σ (z, ɛ) ɛ d/2 L cl σ,d V (x) (z + gd4 )) σ+d/2 ɛ dx. M Harrell-Stubbe TAMS to appear

61 Extension of Reilly s inequality (with El Soufi-Ilias) λ k C(d, k) h 2 ( )

62 Take-away messages #3 1. On manifolds, sum rules involve mean curvature in an explicit way 2. Sharp for spheres where potential = g h Semiclassical inequality requires a partial differential inequality. 4. Each eigenvalue dominated by mean curvature.

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 L-T onedimensional for:

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

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

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

71 Quantum graphs: 1. But not balloons! (A.k.a. tadpoles, or...) Put a soliton potential on the loop: V = φ = cosh(al) cosh(ax) 2a2 cosh 2 (ax) χ loop resp. e ax

72 Quantum graphs: 1. But not balloons! (A.k.a. tadpoles, or...) Put a soliton potential on the loop: V = 2a2 cosh 2 (ax) χ loop φ = cosh(al) resp. e ax cosh(ax) cosh(ax) λ 1 = a 2 solves a transcendental equation, but is exactly determined! λ 1 σ R V σ+1/2

73 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 ) =

74 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?

75 1. Trees. Quantum graphs

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

77 Quantum graphs P But not balloons! L=2 L= λ 2 λ 1 = ( π arctan(1/ ) (2)) arctan(1/ (2)) =16.8

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

79 Why?

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.

81 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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88 When does a quadratic inequality hold? If the graph can be covered by a family of transits where on each edge G = P cst, and for each edge there is some G where this constant is not 0, then j (z λ j ) + 4ɛ a max (z λ j ) + φ a j 2 0. min ( )

89 When does a quadratic inequality hold? Conjecture: This is possible unless the graph can be disconnected from all leaves by removal of one point, or contains a Wheatstone bridge

90 Take-away messages #4 1. On quantum graphs, sum rules reflect the topology. 2. The QG is spectrally one-dimensional if the graph can be covered uniformly by a family of functions that resemble coordinate functions as much as possible. 3. This is not always possible: Connected with a question of classical circuit theory. 4. Full understanding of role of topology is open.

91 Articles related to this seminar S. Demirel and E.M. Harrell, Rev.Math. Physics, to appear. E.M. Harrell and J. Stubbe, On Trace Identities and Universal Eigenvalue Estimates for Some Partial Differential Operators, Trans. Amer. Math. Soc. 349 (1997) E.M. Harrell, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Communications PDE, 2007 A. El Soufi, E.M. Harrell, and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans AMS,2009. E.M. Harrell and L. Hermi, Differential inequalities for Riesz means and Weyltype bounds for eigenvalues, J. Funct. Analysis E.M. Harrell and L. Hermi, On Riesz Means of Eigenvalues, preprint E.M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators, preprint E.M. Harrell and S. Yildirim Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, J. Funct. Analysis E.M. Harrell and J. Stubbe, Trace identities for eigenvalues, with applications to periodic Schrödinger operators and to the geometry of numbers, Trans. AMS, to appear.

92 Take-away messages #N THE END Arne