On a universal limit conjecture for the nodal count statistics of quantum graphs

Speaker: 

Lior Alon

Institution: 

Technion

Time: 

Thursday, October 17, 2019 - 2:00pm to 3:00pm

Location: 

RH 340 P

 

 
 Understanding the statistical properties of Laplacian eigenfunctions in general and their nodal sets in particular, have an important role in the field of spectral geometry, and interest both mathematicians and physicists. A quantum graph is a system of a metric graph with a self-adjoint Schrodinger operator. It was proven for quantum graphs that the number of points on
 which each eigenfunction vanish (also known as the nodal count) is
 bounded away from the spectral position of the eigenvalue by the first Betti number of the graph. A remarkable result by Berkolaiko and Weyand showed that the nodal surplus is equal to a magnetic stability index of the corresponding eigenvalue. A similar result for discrete graphs holds as well proved first by Berkoliako and later by Colin deVerdiere.
 
 Both from the nodal count point of view and the magnetic point of view, it is interesting to consider the distribution of these indices over the spectrum. In our work, we show that such a density exists and defines a nodal count distribution. Moreover, this distribution is symmetric, which allows deducing the topology of a graph from its nodal count. Although for general graphs we can not a priori calculate the nodal count distribution, we proved that a certain family of graphs will have a binomial distribution. As a corollary, given any sequence of graphs from that family with an increasing number of cycles, the sequence of nodal count distributions, properly normalized, will converge to a normal distribution. 
A numerical study indicates that this property might be universal and led us to state the following conjecture. For every sequence of graphs with an increasing number of cycles, the corresponding sequence of properly normalized nodal count distributions will converge to a normal distribution. 
 In my talk, I will present our latest results extending the number
 of families of graphs for which we can prove the conjecture.
 
This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M)

Embedded Eigenvalues for multilayer quantum Graph Graphene

Speaker: 

Lee Fisher

Institution: 

LSU

Time: 

Friday, September 13, 2019 - 2:00pm to 3:00pm

Location: 

RH 306

 

 By judiciously constructing local defects in graph models of multi-layer graphene, bound states can be constructed at energies that lie within the continuous spectrum of the associated Schrödinger operator. The layers may be stacked in AA or AB fashion. A necessary condition for this construction is the reducibility of the Fermi surface for the multi-layer structure. This is achieved due to a special reduction of the complex dispersion relation to a function of a single polynomial "composite" function of the quasimomenta.
This is joint work with Wei Li and Stephen Shipman at LSU.

Nonstationary low-dimensional dynamics

Speaker: 

Victor Kleptsyn

Institution: 

IRM de Rennes

Time: 

Thursday, September 26, 2019 - 2:00pm to 3:00pm

Location: 

RH 340 P

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.

An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products,
which is also of independent interest. 

This is a joint project with A.Gorodetski.

Quantitative almost reducibility of quasiperiodic cocycles

Speaker: 

Lingrui Ge

Institution: 

Nanjing University

Time: 

Thursday, August 29, 2019 - 2:00pm to 3:00pm

Location: 

RH 306

 

Abstract: In this talk, I will give a brief introduction to several popular topics in the spectral theory of quasi-periodic Schrodinger operators. I will then talk about several sharp results we get recently on these topics (especially for almost Mathieu operator). Our results are based on quantitative almost reducibility, a method originally proposed by Dinaburg and Sinai. Finally I will explain the key points in developing and refining this method to get optimal results.

 

A class of Schrodinger operators with convergent perturbation series

Speaker: 

ILYA KACHKOVSKIY

Institution: 

Michigan State

Time: 

Monday, August 12, 2019 - 2:00pm

Location: 

RH 306

Abstract: Rayleigh--Schrodinger perturbation series is one of the main tools of analyzing eigenvalues and eigenvectors of operators in quantum mechanics. The first part of the talk is expository: I will explain a way of representing all terms of the series in terms of graphs with certain structure (similar representations appear in physical literature in various forms). The second part of talk is based on joint work in progress with L. Parnovski and R. Shterenberg. We show that, for a class of lattice Schrodinger operators with unbounded quasiperiodic potentials, one can establish convergence of these series (which is surprising because the eigenvalues are not isolated). The proof is based on the careful analysis of the graphical structure of terms in order to identify cancellations between terms that contain small denominators. The result implies Anderson localization for a class of Maryland-type models on higher-dimensional lattices.

On the kinetic description of the long-time behavior of dispersive PDE

Speaker: 

Zaher Hani

Institution: 

University of Michigan

Time: 

Wednesday, July 3, 2019 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the "wave kinetic equation". This is the wave-analog of Boltzmann's equation for particle collisions. We shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and explore some of its effective long-time behaviors at time scales that are shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).

Regularity of the Density of States for Random Schrodinger Operators

Speaker: 

Krishna Maddaly

Institution: 

Ashoka University, India

Time: 

Tuesday, June 25, 2019 - 2:00pm to 2:59pm

Host: 

Location: 

RH 306

In this talk, a joint work with Dhriti Dolai and Anish Mallick, I will present a proof of smoothness of the density of states for Random Schrodinger operators in any dimension.  We show that the integrated density of states is almost as smooth as the single site distribution of the random potential, in the region of exponential localisation.  The proof relies on the fractional moment bounds on the operator kernels in such energy region.

Our proof also gives a part of the results for the Anderson type models proved by Abel Klein and collaborators more than thirty years ago.

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