Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse...

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Universal patterns in spectra of differential operators Copyright 2008 by Evans M. Harrell II.

Transcript of Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse...

Page 1: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Universal patterns in spectra of differential operators

Copyright 2008 by Evans M. Harrell II.

Page 2: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

This lecture will not involve snakes, machetes, or other dangerous objects.

Page 3: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Laplace, Laplace-Beltrami, and Schrödinger

H = T + V(x)

-

Page 4: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Laplace, Laplace-Beltrami, and Schrödinger

H uk = λk uk

Page 5: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Musicians’ vibrating membranes

Inuit

Page 6: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Musicians’ vibrating membranes

Inuit

Page 7: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Musicians’ vibrating membranes

Inuit

Page 8: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Musicians’ vibrating membranes

Inuit

West Africa

Page 9: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Musicians’ vibrating membranes

Inuit

West Africa

USA

Page 10: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Physicists’ vibrating membranes

Sapoval 1989

Page 11: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Physicists’ vibrating membranes

Sapoval 1989

Page 12: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Physicists’ vibrating membranes

Sapoval 1989

Page 13: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Mathematicians’ vibrating membranes

Wolfram MathWorld, following Gordon, Webb, Wolpert. (“Bilby and Hawk”)

Page 14: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Mathematicians’ vibrating membranes

Wolfram MathWorld, following Gordon, Webb, Wolpert. (“Bilby and Hawk”)

Page 15: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Quantum waveguides - same math (sort of), different physics

Ljubljana Max Planck Inst.

Page 16: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of
Page 17: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

There’s something Strang about this column.

Page 18: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

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

Page 19: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

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

Page 20: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

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

Page 21: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

But according to the Ashbaugh-Benguria Theorem, λ2/ λ1 ≤ 2.5387…!

(In 2 D)

Page 22: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Aha!

Page 23: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Universal spectral patterns

 The top of the spectrum is controlled by the bottom!

 Example: Two classic results relate the spectrum to the volume of the region. For definiteness, consider

H= -∇2 , Dirichlet BC (i.e., functions vanish on ∂Ω).

Page 24: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

A simple universal spectral pattern (for the Dirichlet Laplacian)

 The Weyl law (B=std ball): λk (Ω) ∼ Cd (|B|k /|Ω|)2/d as k→∞.  Faber-Krahn: λ1(Ω) ≥ λ1(B) (|B|/|Ω|)2/d .  Therefore, universally, lim (λk/λ1 k2/d) ≤ Cd/λ1(B).  Equality only for the ball.

Page 25: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Statistics of spectra

For Laplacian on a bounded domain,

Harrell-Stubbe TAMS 1996

Page 26: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Statistics of spectra

A reverse Cauchy inequality:

The variance is dominated by the square of the mean.

Page 27: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Statistics of spectra

Dk :=

Page 28: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Means of ratios:

Harrell-Hermi preprint 2007

Page 29: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Reverse arithmetric-geometric mean inequalities:

For Laplacian on a bounded domain, 0 < r ≤ 3,

Harrell-Stubbe in prep.

Page 30: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Surfaces and other “immersed manifolds”

El Soufi-Harrell-Ilias preprint 2007

Page 31: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Surfaces and other “immersed manifolds”

Harrell CPDE 2007

For a Schrödinger operator T+V(x) on a closed hypersurface,

λ2 – λ1 ≤

El Soufi-Harrell-Ilias preprint 2007

Statistical bounds for eigenvalue gaps, which are all sharp in the case V = g h2, standard sphere.

Page 32: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Surfaces and other “immersed manifolds”

El Soufi-Harrell-Ilias preprint 2007

For Laplace-Beltrami:

Page 33: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Sum rules and Yang-type inequalities

Page 34: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

“Universal” constraints on eigenvalues

 H. Weyl, 1910, Laplace operator λn ~ n2/d  W. Kuhn, F. Reiche, W. Thomas, 1925, sum

rules for energy eigenvalues of Schrödinger operators

 L. Payne, G. Pólya, H. Weinberger, 1956: gaps between eigenvalues of Laplacian controlled by sums of lower eigenvalues:

 E. Lieb and W. Thirring, 1977, P. Li, S.T. Yau, 1983, sums of eigenvalues, powers of eigenvalues.

 Hile-Protter, 1980, stronger but more complicated analogue of PPW

Page 35: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

“Universal” constraints on eigenvalues

 Ashbaugh-Benguria 1991, isoperimetric conjecture of PPW proved.

 H. Yang 1991-5, unpublished, complicated formulae like PPW, respecting Weyl asymptotics.

 Harrell, Harrell-Michel, Harrell-Stubbe, 1993-present, commutators.

 Hermi PhD thesis, articles by Levitin and Parnovsky.

Page 36: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

PPW – the grandfather of many universal spectral patterns:

 All eigenvalues are controlled by the ones lower down! (Payne-Pólya-Weinberger, 1956)

 Not so far off Ashbaugh-Benguria (In 2D, λ2/λ2 ≤ 3 rather than 2.5387….

Page 37: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Universal spectral patterns

 Numerous extensions by same method: Cook up a trial function for min-max by multiplying u1, … uk by a coordinate function xα. For example:

Hile-Protter 1980 Compare to PPW:

Page 38: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Universal spectral patterns

 Numerous extensions by same method: Cook up a trial function for min-max by multiplying u1, … uk by a coordinate function xα.

 Before 1993, these were all lousy for large k, as we knew by Weyl.

Page 39: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

H.C. Yang, unpublished preprint, 1991,3,5

Page 40: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

H.C. Yang, unpublished preprint, 1991,3,5

Some people will do anything for an eigenvalue!

Page 41: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

H.C. Yang, unpublished preprint, 1991,3,5

If you estimate with the Weyl law λj (Ω) ∼ j2/d , the two sides agree with the correct constant.

Page 42: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

What is the underlying idea behind these universal spectral relations and how can you

extract consequences you can understand?

Page 43: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Commutators of operators

 [H, G] := HG - GH  [H, G] uk = (H - λk) G uk  If H=H*, <uj,[H, G] uk> = (λj - λk) <uj,Guk>

Page 44: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Commutators of operators

 [G, [H, G]] = 2 GHG - G2H - HG2  Etc., etc. Try this one:

Page 45: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Commutators of operators

 For (flat) Schrödinger operators, G=xα,  [H, G] = - 2 ∂/∂xα  [G, [H, G]] = 2

Page 46: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Commutators of operators

Page 47: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Commutators of operators

Compares with Hile-Protter:

Page 48: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Commutators of operators

Variant: If H is the Laplace-Beltrami operator + V(x), there is an additional term involving mean curvature.

Page 49: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

What is the underlying idea behind these universal spectral relations, and how can you

extract consequences you can understand?

 The counting function, N(z) := #(λk ≤ z)

Page 50: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

What is the underlying idea behind these universal spectral relations, and how can you

extract consequences you can understand?

 The counting function, N(z) := #(λk ≤ z)  Integrals of the counting function,

known as Riesz means (Safarov, Laptev, etc.):

 Chandrasekharan and Minakshisundaram, 1952

Page 51: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Trace identities imply differential inequalities

Page 52: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Trace identities imply differential inequalities

Page 53: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Trace identities imply differential inequalities

It follows that R2(z)/z2+d/2 is an increasing function.

Page 54: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Understandable eigenvalue bounds from inequalities on Rρ(z):

 Legendre transform – gives the ratio bounds

 Laplace transform – gives a lower bound on Z(t) = Tr(exp(-Ht)).

 The latter implies a lower bound on the spectral zeta function (through the Mellin transform)

Page 55: Patterns in Spectra - Mathphysics.comHarrell-Stubbe TAMS 1996. Statistics of spectra A reverse Cauchy inequality: The variance is dominated by the square of the mean. Statistics of

Summary

 Eigenvalues fall into universal patterns, with distinct statistical signatures

 Spectra contolled by “kinetic energy” and geometry

 Underlying idea: Identities for commutators

 Extract information about λk by differential inequalities, transforms.