Sums of eigenvalues and what they reveal about graphs and manifolds

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Emory Lloyd
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3 Abstract ª Fifty years ago this year, Marc Kac wrote an inspirational article, "Can one hear the shape of a drum?", in which he pondered whether the sounds of the normal modes of vibration of a membrane determine its shape. The frequencies of these normal modes are determined by the eigenvalues of a certain differential equation, and the subject known as spectral geometry, where we try to find out about a shape by "listening" to how it vibrates, has grown since Kac's time to a very active part of modern mathematical analysis. In this lecture I'll begin with a few of the highlights of this subject. Then I'll present a recently discovered variational principle for sums of eigenvalues, and show how it can be used to "hear" the shapes of graphs (networks) and manifolds (regions and surfaces).

M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 1966.

4 M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, ª - Δ u = (ω/c) 2 u =: λ u, with Dirichlet-type boundary conditions.

5 Borg in 1946 and then the school of Gel fand had earlier considered the question of whether one could hear the density of a guitar string, ª - uʹʹ + q(x) u = λ u.

6 Borg in 1946 and then the school of Gel fand had earlier considered the question of whether one could hear the density of a guitar string, ª - uʹʹ + q(x) u = λ u, but they failed to find as charming a catch phrase.

7 Well, can one hear the shape of a drum?

8 Well, can one hear the shape of a drum? Circular ones, sure. No other drum has the same spectrum as the circular drum of a given area. So, the answer is, at least sometimes.

9 Well, can one hear the shape of a drum? H. Urakawa,1982 いいえ!

10 Well, can one hear the shape of a drum? H. Urakawa,1982 いいえ! (At least in 8 or more dimensions)

11 Well, can one hear the shape of a drum? Gordon, Webb, and Wolpert, 1991

12 Well, can one hear the shape of a drum? Gordon, Webb, and Wolpert, They used ideas of Sunada 1985.

13 Some things are audible ª You can hear the area of the drum, by the Weyl asymptotics: ª For the drum problem λ k ~ C d (Vol(Ω)/k) 2/d. (A mathematician s drum can be d- dimensional, and even a curved manifold.)

14 Some things are audible ª You can hear the area of the drum, by the Weyl asymptotics: ª For the drum problem λ k ~ C d (Vol(Ω)/k) 2/d. ª Notice that in addition to the volume,we can hear the dimension.

15 Well, can one hear the shape of a drum? Extreme cases are often unique. For example, what drum (of a given area or in a given category, such as convex) maximizes or minimizes λ 1 or some other λ k?

16 To extremists, things tend to sound simple

17 Is the extremum always a union of round shapes?

18 Is the extremum always a union of round shapes?

19 We can hear the shape of a sufficiently smooth drum ª Zelditch, Ann. Math., Given an analytic boundary, no holes, a line of reflection, and a condition on the closed billiard trajectories,, yes, the drum is uniquely determined by the spectrum!

20 What does the average eigenvalue know?

21 What does the average eigenvalue know? And (gulp!) what am I going to need to know to follow along, now that we are wading into analysis?

22 Mathematical background 1. You can find eigenvalues variationally. 2. As some stage I ll use the Fourier transform, and the main thing you need is the Parseval relation. I may also mention tight frames, which enjoy a similar property. 3. It could be helpful to have seen the Laplace transform before, at least for intuition. 4. PDEs often have Dirichlet or Neumann BC

23 Variational eigenvalues The eigenvalues of a self-adjoint operator (like a symmetric matrix) are are determined by a min-max procedure. The lowest one satisfies the Rayleigh-Ritz inequality, while the other ones satisfy a min-max formula involving orthogonalization.

24 Fourier transform The Fourier transform has various normalizations; the one I ll use is

25 Fourier transform Its completeness, or Parseval relation (actually known to Fourier) reads:

26 Laplace transform

27 Boundary conditions Dirichlet type BC fix the value of the unknown function on the boundary (0 for eigenfunctions), while Neumann BC fix the outward gradient. These are physically natural and guarantee selfadjointness, which give us real eigenvalues and a completeness relation. The weak forms of these BC are nice: E(u) = On appropriate spaces (H 0 and H 1 0 ) Z ru 2

28 Averages of eigenvalues ª Suppose that you know about (say, upper or lower bounds). What else do you know?

29 Averages of eigenvalues ª It is a small step from sums to Riesz means, In particular,

30 Averages of eigenvalues ª Classical transforms can more directly convert bounds on Riesz means (pretty much equivalent to sums) into bounds on a class of functions including the spectral partition and zeta functions.

31 Averages of eigenvalues ª With the Laplace transform,

32 Averages of eigenvalues ª With the Laplace transform, so knowing about sums means knowing about the spectral partition function, which in turn connects to the spectral zeta function.

33 Karamata s theorem A theorem of Karamata provides similar information without passing through Riesz means and transforms. (Karamata is a strengthening of Jensen if you have additional information about ordering and domination.)

34 Averages of eigenvalues ª With Karamata s theorem, a bound of the form implies that for decreasing convex functions (e.g., the spectral partition and zeta functions).

35 Spectral averages, geometry, and dimensionality For Laplacians (DBC or NBC): ª Weyl law:

36 Spectral averages, geometry, and dimensionality For Laplacians (DBC or NBC): ª Weyl law: ª Pólya conjectured in the 60 s that this is a strict lower bound for DBC, and proved it for tiling domains, and a similar result is valid in the other sense for NBC, but this holy grail has still not been proved in full generality.

37 Spectral averages, geometry, and dimensionality For Laplacians (DBC): ª Weyl law: ª However, averaging helps:

38 Spectral averages, geometry, and dimensionality For Laplacians (DBC): ª Weyl law: ª Berezin-Li-Yau

39 Spectral averages, geometry, and dimensionality For Laplacians (DBC): ª Weyl law: ª Berezin-Li-Yau ª Harrell-Hermi for all k > j JFA 08 cst improved by Harrell-Stubbe, 2011.

40 Spectral averages, geometry, and dimensionality ª A rather large industry exists in this area, especially in the School of Laptev Weidl, Frank, Geisinger, Kovarik, and others, and through some insights of Melas. ª Nice interplay of ª Semiclassical analysis ª Geometry ª Functional analysis ª Applications to many-body physics.

41 Variational bounds on sums In 1992 Pawel Kröger found a variational argument for the Neumann counterpart to Berezin-Li-Yau, i.e. a Weyl-sharp upper bounds on sums of the eigenvalues of the Neumann Laplacian. BLY: Kröger:

Berezin-Li-Yau, i.e. a Weyl-sharp upper bounds on sums

42 Variational bounds on sums In 1992 Pawel Kröger found a variational argument for the Neumann counterpart to Berezin-Li-Yau, i.e. a Weyl-sharp upper bounds on sums of the eigenvalues of the Neumann Laplacian. BLY: Kröger:

43 A new hammer in search of nails

44 A new hammer in search of nails (It s just an average hammer.)

45 The average hammer where k Harrell-Stubbe LAA, 2014

46 The average hammer Averages within averages! Harrell-Stubbe LAA, 2014

47 The averaged variational principle for sums Theorem 3.1 Consider a self-adjoint operator M on a Hilbert space H, with ordered, entirely discrete spectrum 1 <µ 0 apple µ 1 apple... and corresponding normalized eigenvectors { (`)}. Let f be a family of vectors in Q(M) indexed by a variable ranging over a measure space (M,, ). Suppose that M 0 is a subset of M. Then for any eigenvalue µ k of M, µ k ZM 0 hf,f i d apple Z M 0 hhf,f id k 1 X j=0 k 1 X j=0 Z µ j Z M M hf, (j) i 2 d! hf, (j) i 2 d, (3.2) provided that the integrals converge. Harrell-Stubbe LAA, 2014

48 Recent applications of the averaged variational principle: 1. Harrell-Stubbe, LAA 2014: Weyl-type upper bounds on sums of eigenvalues of (discrete) graph Laplacians and related operators. 2. El Soufi-Harrell-Ilias-Stubbe, J. Spectral Theory, to appear: Semiclassically sharp Neumann bounds for a large family of 2 nd order PDEs. 3. Harrell-Dever, stuff on blackboards: Quantum graphs. 4. Harrell-Stubbe, semiclassically sharp upper bound for Dirichlet. (Related to a bound of Lieb-Loss.)

49 Combinatorial graphs A graph connects n vertices with edges as specified by an adjacency matrix A, with a ij = 1 when i and j are connected, otherwise 0. The graph is not a priori living in Euclidean space. But it might be! Clearly it could at worst be embedded in R d-1, but what s the minimal dimension?

50 Combinatorial graphs We could debate what we mean by embedding, and our decision is not the one that is most usual in graph theory, which is just whether the graph can be drawn in the plane or not.

51 Combinatorial graphs Our choice is whether the graph is a subset of a regular d-dimensional lattice (possibly including some diagonals).

52 More vs less efficient embeddings of a 1D graph.

53 Combinatorial graphs A graph connects n vertices with edges as specified by an adjacency matrix A, with a ij = 1 when i and j are connected, otherwise 0. The graph is not a priori living in Euclidean space. But it might be! Clearly it could at worst be embedded in R d-1, but what s the minimal dimension? Can you figure that out by listening to the eigenvalues?

54 Combinatorial graphs A graph connects n vertices with edges as specified by an adjacency matrix A, with a ij = 1 when i and j are connected, otherwise 0. The graph is not a priori living in Euclidean space. There is a natural Laplacian on the graph, with non-negative spectrum, which has been heavily studied, but not much with this question in mind. Harrell-Stubbe LAA, 2014

55 The graph Laplacian The graph Laplacian is a matrix that compares values of a function at a vertex with the average of its values at the neighbors. H := -Δ := Deg A, where Deg := diag(d v ), d v := # neighbors of v. Its weak form is:

56 How complex is a branch of mathematics considered as an embedded graph? Undergraduate research project of Philippe Laban See (under development)

57 The mathematics pages of Wikipedia Category theory stubs Applied mathematics stubs Mathematical analysis stubs Algebra stubs Mathematician stubs Mathematics literature stubs Cryptography stubs Mathematical tables Mathematical software Mathematical markup languages Calculators Formal methods terminology Dimension Mathematical theorems Mathematical problems Probability and statistics Numerical analysis Mathematical modeling Mathematical tools Mathematical relations Mathematical proofs Mathematical principles Number theory stubs Mathematical logic stubs Mathematics competition stubs Geometry stubs Combinatorics stubs Algorithms Applied mathematics Number stubs Mathematics journal stubs Theories of deduction Philosophy of computer science Mathematics paradoxes Philosophers of mathematics Structuralism (philosophy of mathematics) Topology stubs Statistics stubs Probability stubs Mathematical objects Mathematical logic Mathematics and mysticism Mathematics stubs Mathematical tools Mathematical terminology Pseudomathematics Mathematical problem solving Numeral systems Mathematical typefaces Philosophy of mathematics Z notation Mathematical symbols Coordinate systems Mathematical markup languages Mathematical relations Basic concepts in set theory Mathematical notation Mathematical principles Mathematical structures Majority Mathematical objects Mathematical concepts Algorithms Mathematical axioms Mathematics timelines History of statistics Actuarial science Mathematics Mathematical problems Computational mathematics Mathematics manuscripts Computational science Cryptography History of mathematics History of logic Cybernetics Mathematical examples History of mathematics journals Mathematical economics Applied mathematics Mathematical finance Mathematical and theoretical biology Mathematical chemistry Mathematical physics Mathematical psychology Mathematics in medicine Mathematics of music Operations research Probability theory Signal processing Theoretical computer science Applied mathematics stubs Mathematics and culture Mathematics by culture Mathematics awards Elementary mathematics Fields of mathematics Mathematics fiction books Mathematics competitions Mathematics conferences Documentary television series about mathematics Mathematics education M. C. Escher Ethnomathematicians Fermat%27s Last Theorem Films about mathematics Mathematical humor Mathematics and art Mathematics and mysticism Mathematics of music Mathematics-related topics in popular cultur Mathematics organizations Mathematical problems Recreational mathematics Square One Television Mathematics websites Elementary algebra Elementary arithmetic Elementary geometry Elementary number theory Algebra Mathematical analysis Applied mathematics Arithmetic Calculus Combinatorics Computational mathematics Discrete mathematics Dynamical systems Elementary mathematics Experimental mathematics Foundations of mathematics Game theory Geometry Graph theory Mathematical logic Mathematics of infinitesimals Number theory Order theory Probability and statistics Recreational mathematics Representation theory Topology Mathematics by culture Mathematics by period History of algebra History of calculus History of computer science History of geometry Hilbert%27s problems Historians of mathematics Historiography of mathematics

58 The adjacency matrix for PDEs in wikipedia

59 The adjacency matrix for PDEs in wikipedia

60 The adjacency matrix for Analysis

61 The adjacency matrix for Graph Theory

62 Connecting the spectrum of a graph and its embedding dimension We can mimic the argument for the Kröger-type bounds by considering test functions of the form exp(ip x), where x is restricted to integer vectors. Is there a Parseval relation? Yes Is there a nice expression for the energy form? Yes

63 Connecting the spectrum of a graph and its embedding dimension

64 Connecting the spectrum of a graph and its embedding dimension

65 Dimension and complexity This is the graph of wikipedia PDE pages. How many independent kinds of information ( dimensions ) are there?

66 Dimension and complexity You might not see it visually, but the spectrum says that this is 3D!

67 Another interesting question: Can you distinguish dimensions on different scales?

68 Another interesting question: Can you distinguish dimensions on different scales? 網

69 Variational bounds on graph spectra Another way to apply the averaged variational principle to graphs is to let M be the set of pairs of vertices. The reason is that the complete graph has a tight frame of nontrivial eigenfunctions consisting of functions equal to 1 on one vertex, -1 on a second, and 0 everywhere else. Let these functions be h z, where z is a vertex pair.

70 Variational bounds on graph spectra Two facts are easily seen: 1. 2.

71 Variational bounds on graph spectra From the averaged variational principle,

72 Proof of the AVP

73 Proof

74 Proof

75 Proof

76 How to use the averaged variational principle to get sharp results?

77 How to use the averaged variational principle to get sharp results?

78 How to use the averaged variational principle to get sharp results?

79 How to use the averaged variational principle to get sharp results? Ans: If is large enough that then

80 How to use the averaged variational principle to get sharp results?

81 Example: Recover Kröger s result Our theorem says that IF that is sufficiently big Then we have an upper bound on a sum involving eigenvalues. For trial functions we take the Fourier exponential functions, and we equate ζ with the dual variable p.

82 Example: Recover Kröger s result With the Parseval identity, IF then Choosing as a ball of radius R in p-space, a simple calculation gives Kröger.

83 The mother of all PDEs with Neumann BC: We can find sharp upper bounds for sums of eigenvalues of expressions defined in a variational quadratic form as follows: Where Ω is a domain in a homogeneous space, which has been conformally transformed in an arbitrary way. Weak Neumann conditions correspond to test functions in the restriction of H 01 (R d ) to Ω. (Evans and Edmunds)

84 The mother of all PDEs with Neumann BC: The key is that we can find adapted Fourier functions where the energy is computable, and a Parseval identity (or inequality) to allow the application of the averaged variational principle.

85 A simpler example: Quantum graphs Here the edges of the graph matter and carry a differential equation. In the absence of a potential energy, the weak form of the quantum graph operator is just the usual integral of the square of the derivative:

86 Bounds on sums for quantum graphs

87 Bounds on sums for quantum graphs For today we ll set V=0, and avoid the temptation to introduce other complications.

88 Bounds on sums for quantum graphs For today we ll set V=0, and avoid the temptation to introduce other complications. Well, other than some general remarks.

89 Bounds on sums for quantum graphs To get the machine running, we d like a set of trial functions which have a nice relation to the operator and a completeness relation, so the Fourier exponentials again come to mind.

90 An adapted Fourier transform

91 An adapted Fourier transform

92 An adapted Fourier transform

93 Bounds on sums for quantum graphs

94 Bounds on sums for quantum graphs

95 Bounds on sums for quantum graphs

96 Bounds on sums for quantum graphs

97 Bounds on sums for quantum graphs

98 Bounds on sums for quantum graphs

99 Bounds on individual eigenvalues Recall that we threw something out in the derivation. But you don t have to throw that part out, and using the earlier result, you can circle back and get bounds on individual eigenvalues. With some work

100 Bounds on sums for quantum graphs This result is in the form that applies to the case of Euclidean domains, where m k is the Weyl expression, but a similar result works for all of our applications, including quantum graphs. (Harrell-Stubbe, unpublished)

101 PDEs on homogeneous spaces

102 PDEs on homogeneous spaces A homogeneous space is a manifold M with a continuous symmetry group of isomorphisms M M. Canonical examples: R d, S d, H d.

103 The weak form of PDEs with Neumann BC We can find sharp upper bounds for sums of eigenvalues of expressions defined in a variational quadratic form as follows: Where Ω is a domain in a homogeneous space, which has been conformally transformed in an arbitrary way. Weak Neumann conditions correspond to test functions in the restriction of H 01 (R d ) to Ω. (Evans and Edmunds)

104

Extensions to other homogeneous spaces Recently, with El Soufi and Ilias we derived sharp inequalities for Neumann Laplacian eigenvalues on subdomains of

105 Extensions to other homogeneous spaces Recently, with El Soufi and Ilias we derived sharp inequalities for Neumann Laplacian eigenvalues on subdomains of homogeneous spaces, including extensions of Kröger for domains conformal to R d, and analogues on compact such as with implications for Z(t). Cf. Strichartz, 1996.

106 Proof

107 By the theorem,

108 Phase-space bounds

109 For a better result with a potential, Use coherent states (like wavelets): resp. Cf. Lieb-Loss

110 Phase space estimates with V(x) ª The intuition in physics is that each eigenfunction fills a certain volume in phase space, Thus the number of energy levels (eigenvalues) Λ is proportional to the volume of the region in phase space for which

111 For domains conformal to Euclidean sets, we take with which we find a Parseval relation and a reasonable energy calculation. (for details, see the paper).

112 THE END

What does the average eigenvalue know?

What does the average eigenvalue know? Evans Harrell Georgia Tech www.math.gatech.edu/~harrell CWRU March, 2017 Copyright 2017 by Evans M. Harrell II. What does the average eigenvalue know about geometry,