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Spectral Theory and Its Applications

October 8-10, 2014 at University Bordeaux

Programme




Wednesday October 8:

13h30 Yves Colin de Verdière: Quantum ergodicity for sub-Riemannian Laplacianspdf
14h30 Batu Güneysu: Scattering theory of the Hodge Laplacian on noncompact manifolds pdf
15h30 Coffee Break
16h00 Matthias Keller: Intrinsic metrics on graphs
17h00 Bobo Hua: A Variant of Davies Lemma on Graphspdf



Thursday October 9:

8h30 Nalini Anantharaman: Quantum ergodicity on large regular graphs
9h30 Coffee Break
10h00 Fabricio Macia: Energy decay for the damped wave equation on a regular graph
11h00 Justin Salez: Atoms in the limiting spectrum of diluted random graphspdf
12h00 Lunch
14h Laurent Miclo: On hyperboundedness and spectrum of Markov operatorspdf
15h00 Charles Bordenave: Non-backtracking spectrum of random graphspdf
16h00 Coffee Break
16h30 Roman Schubert: Entropy of Eigenfunctions on Quantum Graphspdf
20h00 Restaurant Le Karma



Friday October 10:

8h30 Jacob Christiansen: Orthogonal polynomials on infinite gap setspdf
9h30 Coffee Break
10h00 Sergey Naboko: The fnite section method for dissipative Jacobi and Schrodinger operators
11h00 Sergey Simonov: Kac theorem for Schroedinger operators on star-graphs



Abstracts:

Yves Colin de Verdière: Quantum ergodicity for sub-Riemannian Laplacians

This is joint work in progress with Luc Hillairet (Orléans) & Emmanuel Trélat (Paris 6). A. Shnirelman proved in 1974 the following Theorem: "let $(X,g)$ be a closed Riemannian manifold with ergodic geodesic flow and $(\phi_n,\lambda_n)$ an eigen-decomposition of the Laplace operator. Then there exists a density $1$ sub-sequence $\lambda _{n_j}$ of the sequence $\lambda _n $ so that the sequence of probability measures $|\phi_{n_j}|^2 dx_g $ ($dx_g$ being the Riemannian measure) converges weakly to the measure $dx_g$." I plan to describe extensions of this Theorem to the case of contact 3D sR Laplacians and to discuss possible extensions to other sR geometries.

Batu Güneysu: Scattering theory of the Hodge Laplacian on noncompact manifolds

In this talk, given two Riemannian metrics g and h on a noncompact manifold M, I will explain a very general first order criterion on the deviation of the metrics g and h for the existence and completeness of the wave operators corresponding to the Hodge-Laplacians \Delta_g and \Delta_h (acting on differential forms). This result generalizes earlier results by Müller/Salomonsen and Hempel/Post/Weder away from functions.

Matthias Keller: Intrinsic metrics on graphs

There are various results in Riemannian manifolds which are consequences of the close relationship between the Laplace-Beltrami operator and the Riemannian distance. For graphs many of these results fail to be true if one considers the combinatorial graph distance. Now by using the concept of intrinsic metrics recently introduced in the context of general regular Dirichlet forms by Frank/Lenz/Wingert analogous results can be proven in the context of weighted graphs. This solves various problems that have been open for several years and decades.

Bobo Hua: A Variant of Davies Lemma on Graphs

The classical Davies Lemma gives the double integral estimate of the heat kernel on any complete Riemannian manifold. On a general metric measure space, Coulhon and Sikora proved that this Davies Lemma is equivalent to the finite propagation speed property of the solutions to wave equations. However, Friedman and Tillich gave a counterexample to finite propagation speed property of wave equations on graphs. In this talk, we prove a variant of Davies Lemma on graphs which yields the Gaussian heat kernel upper bounds for the continuous time heat kernel under the curvature dimension condition.

Nalini Anantharaman: Quantum ergodicity on large regular graph

In recent work with E. Le Masson, we have proved a form of delocalization of eigenfunctions of the laplacian on large regular graphs, called "quantum ergodicity". The graphs need not be random, but only need to form a family of expanders and to converge to tree in the sense of Benjamini-Schramm. I will give two proofs of the result, the first is the one that appears in the paper with Le Masson but the second one is probably easier to adapt to the case of "laplacian + potential" or of non-regular graphs.

Fabricio Macia: Energy decay for the damped wave equation on a regular graph

We study the decay rate of the energy of solutions to the damped wave equation on a graph obtained as a regular discretization of a flat torus. We obtain necessary and sufficient conditions for the existence of a uniform decay rate (in terms of the initial data and the number of vertices of the graph). This problem is well understood in the continuous setting in terms of a geometric condition on the support of the damping term (the Geometric Control Condition). The answer to the problem in the discrete setting is a bit more complex, and involves different wave-length scales. Our proofs are based on an analysis of the propagation of the energy density of solutions to the homogeneous wave equation on the graph via two-microlocal semiclassical measures. This is joint work with Hans Christianson (UNC).

Justin Salez: Atoms in the limiting spectrum of diluted random graphs

A decade ago, Khorunzhy, Shcherbina and Vengerovsky established the convergence of the empirical spectral distribution of large Erdos-Renyi random graphs with fixed average degree. Yet, only very little is known about the limiting measure. In particular, Ben Arous asked for the precise location of its atoms (Open Problem 14 of the 2010 AIM Workshop on Random Matrices). In this talk, I will present a complete answer to this question.

Laurent Miclo: On hyperboundedness and spectrum of Markov operators

Consider an ergodic Markov operator $M$ reversible with respect to a probability measure $\mu$ on a general measurable space. We will show that if $M$ is bounded from ${\cal L}^2(\mu)$ to ${\cal L}^p(\mu)$, where $p>2$, then it admits a spectral gap. This result answers positively a conjecture raised by H{\o}egh-Krohn and Simon in a semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee, Gharan and Trevisan. In general there is no quantitative link between hyperboundedness and spectral gap (except in the situation already investigated by Wang), but there is one with another eigenvalue. In addition, the usual Cheeger inequalities will be extended to higher eigenvalues in the compact Riemannian setting.

Charles Bordenave: Non-backtracking spectrum of random graphs

The non-backtracking matrix of a graph is a non-symmetric matrix on the oriented edge of a graph which has interesting algebraic properties and appears notably in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. It has also been used recently in the context of community detection. In this talk, we will study the largest eigenvalues of this matrix for the Erdos-Renyi graph G(n,c/n) and for simple inhomogeneous random graphs (stochastic block model). This is a joint work with Marc Lelarge and Laurent Masoulié.

Roman Schubert: Entropy of Eigenfunctions on Quantum Graphs

We are interested in the distribution of eigenfunctions on quantum graphs, in particular how they depend on the topology of the graph. As a measure for the distribution we consider the entropy; if an eigenfunction has a large entropy it implies that it cannot be concentrated on a small set of edges. We will focus on two classes of graphs, star graphs and regular graphs. For star graphs we show that the average of the entropies of eigenfunctions is small, indicating eigenfunctions which localise on few bonds. In contrast for regular graphs with large girth we show that the entropy of eigenfunctions is large. The strongest estimates we obtain for expanders where we choose the length of the bonds randomly, then we can show that with large probability the entropy is at least half as large as the maximal possible value. This is analogous to the results by Anantharaman and Nonnenmacher on the entropy of quantum limits on Anosov manifolds, and we in particular borrow one of their tools, the entropic uncertainty principle by Maassen and Uffink.

Sergey Naboko: The fnite section method for dissipative Jacobi and Schrodinger operators

The dissipative potential perturbation of a self-adjoint Jacobi matrix or a self-adjoint Schrodinger operator be considered. We show that the discrete spectra of the truncated Jacobi matrices or the restrictions of the differential operators on bounded intervals will approximate any point of the limiting operators spectra ( = finite section method).The talk is based on the common work with M.Marletta ( Cardiff ).

Jacob Christiansen: Orthogonal polynomials on infinite gap sets

The theory of orthogonal polynomials on a compact interval (or a finite union of compact intervals) can be generalized to infinite gap sets of Parreau-Widom type. This notion of regular compact sets includes Cantor sets of positive measure, among others. Central to the theory is the so-called isospectral torus, T, which is a collection of certain reflectionless matrices with the 'right' spectrum. As I'll explain, this is the natural limiting object by deep results of Remling. In spectral theory for orthogonal polynomials, one seeks to relate conditions on the measure of orthogonality to conditions on the recurrence coefficients (which also appear as parameters in the associated Jacobi matrices). In the talk, I'll introduce the Szegö class as a collection of probability measures that contains the isospectral torus as a subset. A long term goal is to relate this class of measures to Jacobi matrices whose parameters are $\ell^p$ perturbations of (points in) T, mainly for $p=1, 2$. Unlike the case of a compact interval, this is far from completely understood. I'll formulate some conjectures and discuss recent progress towards proving them.

Sergey Simonov: Kac theorem for Schroedinger operators on star-graphs

The classic theorem of I. S. Kac says that the singular spectrum of a Schroedinger operator on the whole real line is simple for any potential which is in the limit point case at both infinities. We consider Schroedinger operators on a star-shaped graph with interface (matching) conditions at the interior vertex that correspond to one-dimensional perturbations in the resolvent sense of the Dirichlet decoupling operator. The case of the standard (Kirchhoff) interface condition is included. Note that the Schroedinger operator on the real line can be considered as an operator on a star-graph with two edges (positive and negative half-lines) with the standard interface condition at the origin. We generalize Kac's theorem on the considered class of operators. The multiplicity of their singular spectra is determined by spectral measures of corresponding operators on individual edges with Dirichlet condition at the interior vertex. The talk is based on a joint work with Harald Woracek.