## Offline Mini-Courses

in Spectral Theory and Mathematical Physics

**11 June–22 July, 2021**

Video recordings of the talks are now available online.

These are mini-courses for the EIMI thematic program on Spectral Theory and Mathematical Physics (rescheduled from 2020 due to COVID-19 pandemic). The target audience includes graduate, master and senior bachelor students of any mathematical speciality; senior researchers are also welcome.

The meetings will be held in the Leonhard Euler International Mathematical Institute (Pesochnaya nab. 10, St. Petersburg, Russia).

Please, see the timetable page for the schedule.

Video recordings of the talks are available on the EIMI YouTube channel.

### Marcin Moszyński (Uniwersytet Warszawski, Poland)

*Spectral theory for self-adjoint cyclic operators & its analog for "finitely cyclic" ones with rigorous introduction to matrix measure L*²* spaces*

**19–22 July, 2021**

*I would like to dedicate this mini course to the memory of my friend Sergei Naboko.*

This mini-course is devoted to а rigorous proof of the result, called here "x-Multiplication Unitary Equivalence" (abbr.: xMUE) Тheorem, for *finitely cyclic* self-adjoint operators, which is an analog of the "xMUE" Theorem for self-adjoint cyclic operators (most probably*, it is one of the Stone's results for cyclic – i.e.,"simple spectrum" – operators).

A *finitely cyclic* operator is a generalisation of a cyclic operator: the cyclic ("generating") vector is replaced here by a finite system of vectors (the name an "operator with finite multiplicity of the spectrum" is also used). To do this, we introduce and rigorously develop some foundations of the theory of matrix measure *L*² spaces, and we study spectral properties of multiplication operators (by scalar functions) in such spaces.

Because of the "mini" character of the course, some proofs will be omitted during the lectures, but all the details can be found in the manuscript: Part 1 and Part 2.

**Lecture 1**:

- Introduction
- A detailed proof of the "xMUE" Theorem for self-adjoint cyclic operators

**Lecture 2**:

- Matrix measure and its trace measure, and trace density
- ℂ^
*d*– vectors function space(*L*^*M*) with semi-scalar product, and its "zero-layer"

**Lecture 3**:

- The
(*L*²*M*) Hilbert space (completeness) - The subspaces of simple functions and of "smooth" functions (density)

**Lecture 4**:

- Multiplication operators in
(*L*²*M*) - Self-adjoint finitely cyclic operators

**Lecture 5**:

- The spectral matrix measure for a self-adjoint operator and a system of vectors
- The canonical spectral transformation

**Lecture 6**:

- Vector polynomials and "xMUE" Theorem for self-adjoint finitely cyclic operators

____

* According to the opinion of Sergei N.

### Alexander Its (IUPUI, USA & St. Petersburg University, Russia)

*Nonlinear saddle point method for the cylindrical KdF equation*

**16–18 June, 2021**

We discuss long time asymptotics of the solution to a Cauchy problem for the nonlinear cylindrical KdF equation. Our method is based on an asymptotic analysis of a matrix Riemann-Hilbert problem. We use a well-known version of the nonlinear steepest descent method.

### Elena Zhizhina (Institute for Information Transmission Problems, Russia)

*On periodic homogenization of non-local convolution type operators*

**11–18 June, 2021**

The course will focus on periodic homogenization of parabolic and elliptic problems for integral convolution type operators, it is based on recent results obtained in our joint works with A.Piatnitski.

First, we will consider some models of population dynamics, which can be described in terms of non-local convolution type operators. In particular, we will introduce evolution equations for the dynamics of the first correlation function (the so-called density of population). We will also discuss a number of problems arising in the theory of non-local convolution type operators.

Then we will turn to homogenization of elliptic equations for non-local operators with a symmetric kernel. We will show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. The proof of convergence includes the following steps:

- construction of anzats and deriving the equation on the first corrector;
- construction of the limit diffusive operator;
- justification of the convergence.

Our next goal is homogenization of parabolic problems for operators with a non-symmetric kernel. It will be shown that the homogenization result holds in moving coordinates. We will find the corresponding effective velocity and obtain the limit operator. In the case of small antisymmetric perturbations of a symmetric kernel we will show that the so-called Einstein relation holds.

References:

- A.Piatnitski, E.Zhizhina,
*Periodic homogenization of non-local operators with a convolution type kernel*, SIAM J. Math. Anal. Vol. 49, No. 1, p. 64-81, 2017. - A.Piatnitski, E.Zhizhina,
*Homogenization of biased convolution type operators*, Asymptotic Analysis, 2019, Vol. 115, No. 3-4, p. 241-262, doi:10.3233/ASY-191533; Arxiv: 1812.00027. - Yu.Kondratiev, O.Kutoviy, S.Pirogov, E.Zhizhina,
*Invariant measures for spatial contact model in small dimensions*, Arxiv: 1812.00795, 29 November 2018, Markov Proc. Rel. Fields (2021), to appear.