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

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.

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 9, 17:30-18:30

TBA

June 23, 17:30-18:30

R. Romanov, St.Petersburg State University

TBA

Past talks

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.