Seminar in spectral theory and related topics

These days seminar sessions are zoom meetings each Tuesday. To participate in the seminar, join zoom channel 812-916-426. If you do not know the password, please write an email to Roman Bessonov. To subscribe for seminar anonucements, please visit this page. Some records of talks given on the seminar are avaliable here.

Forcoming talks

July 7 , 15:30-16:30

G. Rozenblum, Chalmers University of Technology and SPbU

Eigenvalue properties of zero order pseudodifferential operators and an application to a 3D elasticity problem.

We consider a zero order pseudodifferential operator on a compact manifold. Suppose that the operator is polynomially compact. Then there may exist sequences of eigenvalues converging to the points of the essential spectrum. We find their asymptotics, by means of reduction to some negative order operators. The motivating example is the Neumann-Poincare (double layer) operator in 3D elastostatics, where the properties of eigenvlues converging to the points of the essential spectrum are studued. The general results enable us to separate the contribution of the elastic properties and of the geometry of the body to the eigenvalue asymptotics. Details can be found in https://arxiv.org/abs/2006.10568

Past talks

June 30, 16:30-17:30

A. Kostenko, University of Vienna

Kirchhoff Laplacians on metric graphs

Laplacians on metric graphs (often called Quantum Graphs) play an important role as an intermediate setting between Laplacians on Riemannian manifolds and Laplacians on graphs and share properties with both. In this talk, we plan to discuss basic spectral properties of Laplacians on metric graphs. The main emphasis will be on graphs with infinitely many vertices and edges, the case which is much less understood.

June 23, 17:30-18:30

R. Romanov, St.Petersburg State University

Discreteness of the spectrum of singular canonical systems.

We are going to provide a criterion for discreteness of the spectrum of canonical systems with one singular end. This criterion simultaneously answers the question of L. de Branges on description of canonical systems corresponding to arbitrary Hermite–Biehler functions. This is a joint work with Harald Woracek (Vienna).

June 16, 15:30-16:30

M. Mitkovski, Clemson University

Uncertainty principles of Paneah-Logvinenko-Sereda type

I will present several new forms of the harmonic analysis uncertainty principle. These new forms can be viewed as a sharpening of the classical Paneah-Logvinenko-Sereda uncertainty principle, in a sense that we impose similar restrictions (sometimes more, sometimes less) on the Fourier support and deduce similar kind of sampling inequalities. I will also present applications of our uncertainty principles to some control and damping problems in linear PDE’s. This is a joint work with W. Green and B. Jaye.

June 9, 17:30-18:30

M. Dorodnyi, EIMI

Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results.

In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$ , we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ , with periodic coefficients depending on $\mathbf{x}/\varepsilon$ . We study the behavior of the operator $\exp(-i \tau \mathcal{A}_\varepsilon)$ for small $\varepsilon$ . It is proved that, as $\varepsilon \to 0$ , this operator converges to $\exp(-i \tau \mathcal{A}^0)$ in the norm of operators acting from the Sobolev space $H^s(\mathbb{R}^d;\mathbb{C}^n)$ (with a suitable $s$ ) to $L_2(\mathbb{R}^d;\mathbb{C}^n)$ . Here $\mathcal{A}^0$ is the effective operator with constant coefficients. We prove sharp-order error estimates and study the question about the sharpness of the results with respect to the norm type, as well as to the dependence of estimates on $\tau$ . The results are applied to study the behavior of the solution $\mathbf{u}_\varepsilon$ of the Cauchy problem for the Schrödinger-type equation $i\partial_{\tau} \mathbf{u}_\varepsilon = \mathcal{A}_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$ .

June 2, 17:30-18:30

V. Peller, St.Petersburg State University

Двойные операторные интегралы и формула следов Лифшица – Крейна

В докладе будут определены двойные операторные интегралы и будут обсуждаться их свойства. Затем мы перейдём к формуле следов Лифшица—Крейна и обсудим, как её можно получить с помощью двойных операторных интегралов. При этом важную роль играет теорема Сёкефальви-Надя и Фойаша об унитарных дилатациях сжатий.

May 26, 17:30-18:30

P. Gubkin, St.Petersburg State University

Asymptotic behavior of Krein system solutions

Krein system is the following system of differential equations:

$\begin{align*}\begin{cases} \frac{\partial}{\partial r}P(r,\lambda) = i\lambda P(r,\lambda) - \ol{a(r)}P_*(r,\lambda), &\quad P(0,\lambda) = 1,\\ \frac{\partial}{\partial r}P_*(r,\lambda) = - a(r) P(r, \lambda), & \quad P_*(0,\lambda) = 1.\end{cases}\end{align*}$

We will discuss why functions $P, P_*$ are usually called continuous analogues of orthogonal polynomials on the unit circle and consider some of their properties mostly related to behavior of $P_*(r,\lambda)$ as $r\to\infty$ .

May 19, 17:30-18:30

V. Sloushch, St.Petersburg State University

Asymptotics of the discrete spectrum appearing in spectral gaps of the discrete Schrodinger operator under decaying sign-definite perturbation

We consider a periodic Schrodinger operator $H$ on discrete periodic graph $\Gamma$ in $\mathbb{R}^{d}$ . The operator $H$ is perturbed by a sign-definite decaying potential $V$ on the graph $\Gamma$ ; $V(x)\sim\vartheta (x/|x|)|x|^{-d/p}$ , $|x|\to\infty$ , $p>0$ . The asymptotics with the large coupling constant of a discrete spectrum of the perturbed operator $H_{\pm}(t):=H\pm tV$ , $t>0$ is investigated.

May 12, 17:30-18:30

T. Weinmann, St.Petersburg State University

Spectral averaging for rank-one perturbations of self-adjoint operators

Let $A$ be a self-adjoint operator on a Hilbert space $H$ and $\varphi\in H$ . We study the properties of the family of rank-one perturbations of $A$ given by $A_\alpha=A+\alpha (\varphi, \cdot)\varphi$ , where $\alpha\in\mathbb{R}$ . In particular, we prove that if $f\in L^2(dx)$ , then $f\in L^2(d\mu^{\varphi}_\alpha)$ for almost every $\alpha$ (here $\mu^{\varphi}_\alpha$ denotes $(\varphi, E_{A_\alpha}(\cdot)\varphi)$ ) and that in this case the spectral averaging formula holds:

$\iint f(x) d\mu^{\varphi}_{\alpha}(x)d\alpha = \int f(x) dx.$

May 5, 17:30-18:30

Yu. Belov, St.Petersburg State University

Localization of zeroes for Cauchy transforms and canonical systems

If discrete measure $\sum_n\mu_n\delta_{t_n}$ is sufficiently small, then the zeroes of Cauchy transform are localized near the $\supp\mu$ . Moreover, it may happen that this holds for any $\nu$ such that $|\nu|<\mu$ . We have found a description of such measures and attraction sets (i.e. subsets of $\supp\mu$ which attract zeroes). We have proved that all attraction sets are ordered by inclusion.

Such measures appear naturaly in the theory of canonical systems of differential equations. They correspond to the canonical systems whose Hamiltonian consists of indivisible intervals accumulating only to the left. Moreover, this correspondence is one-to-one under some additional assumptions. This topic is connected to the problem of density of polynomials and other classical problems in harmonic analysis.

This is joint work with A. Baranov and E. Abakumov.

April 28, 17:30-18:30

E. Korotyaev, St.Petersburg State University

Inverse scattering on half line, new results

We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the S-matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.