## Offline Mini-Courses

in Spectral Theory and Mathematical Physics

**11 June–31 December, 2021**

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.

### Sergey Dobrokhotov (Ishlinsky Institute for Problems in Mechanics RAS & Moscow Institute of Physics and Technology, Russia)

Feynmann–Maslov calculus of functions of noncommuting operators and applications to adiabatic and semiclassical problems

**13 and 20 October, 2021**

At the elementary level, we discuss the basic objects and formulas in the Feynman–Maslov operator theory about functions of non-commuting operators. The main objects here are differential and pseudo-differential operators with a small parameter (h-pseudo-differential operators). As a simple effective application, we consider the use of this theory in adiabatic problems (in particular, in dimension reduction problems). These include vector problems (for example, problems about wave functions in graphene), problems about wave propagation in waveguides (for example, waves in nanotubes), problems for equations with rapidly changing coefficients (averaging methods), etc. As a result of the application of the considered approach (formulated in the form of an algorithm), the initial problems are reduced to simpler problems described by "effective" Hamiltonians or modes, and containing, in particular, dispersion effects. Then one can use the semi-classical approximation to construct asymptotic solutions of the reduced equations.

The first part of the lectures is devoted to elementary definitions and important effective formulas of operator calculus and applications to vector problems (the simplest problems with an "operator-valued symbol"). The second part is devoted to the more complicated adiabatic problems mentioned above.

### Frédéric Klopp (IMJ-PRG, Sorbonne Université, France)

Recent results in localization

**15 and 29 September, 2021**

The mini-course will be devoted to two recent results on localization. The first result deals with one particle localization and is joint work with J.Schenker (https://arxiv.org/abs/2105.13215). It shows that, for quite general random models, while localization cannot be uniform on an non empty open interval of energy, a result which goes back to the 90's, it almost is i.e. only a small fraction of the states does not localize uniformly. The second result deals with many body localization. For a simple one dimensional random Hamiltonian, we show that, in the thermodynamic limit:

- at zero temperature, for a small enough particle density, the ground state of the Hamiltonian of many fermions subjected to this one dimensional random Hamiltonian interacting through a compactly supported repulsive potential exhibits localization;
- at positive temperature, for a small enough chemical potential, the Gibbs state of the grand canonical ensemble for the same Hamiltonian exhibits localization.

Here, in both cases, a state "exhibits localization" if its two particle density matrix decays exponentially off the diagonal.

### Vladimir Nazaikinskii (Ishlinsky Institute for Problems in Mechanics RAS, Russia)

Geometry and semiclassical asymptotics

**8 and 29 September, 2021**

Maslov's canonical operator is one of the most powerful tools for constructing global semiclassical asymptotics for linear differential equations and systems. We will outline the rich geometry underlying the canonical operator (Lagrangian manifolds in the phase space, focal points, caustics, Maslov index, etc.) and explain its up-to-date construction suitable not only for theoretical research but also for the efficient analysis of specific problems using the capabilities of technical computation systems such as Wolfram Mathematica.

### Alexander Pushnitski (King's College London, UK)

Additive and multiplicative Hankel and Toeplitz operators

**8 and 15 September, 2021**

In the first part, I will discuss the classical (additive) Toeplitz and Hankel operators. These are operators whose matrix representations have the form {t(j-k)} for Toeplitz and {h(j+k)} for Hankel (here j,k are non-negative integers). In the second part, I will discuss the multiplicative Toeplitz and Hankel operators; these are operators represented by infinite matrices of the form {t(j/k)} and {h(jk)}, where j,k are natural numbers. It is well known that additive Toeplitz and Hankel operators can be naturally realised as operators on the Hardy space. It turns out that in a similar way the multiplicative Toeplitz and Hankel operators can be realised as operators acting on a certain Hilbert space built from Dirichlet series. I will discuss the general set-up for the theory of these classes of operators and mention some key questions: boundedness, compactness, finite rank property, etc.

### Ilya Kachkovskiy (Michigan State University, USA)

Bourgain’s method of proving Anderson localization for quasiperiodic operators

**28–30 August, 2021**

Anderson localization for Schrödinger operators can be described as the property of having purely point spectrum with exponentially decaying eigenfunctions. For random operators, it was discovered in 1958 by P.Anderson and proved rigorously in many different models. For quasiperiodic operators, perturbative methods of proving Anderson localization based on KAM theory were developed since 1980s. A common property of these methods is that the coupling constant at the potential needs to be large enough, depending on the Diophantine properties of the frequency. In particular, one cannot find a lower bound on the coupling constant that would guarantee Anderson localization for a full measure set of frequencies.

In 1998, S.Jitomirskaya obtained optimal (in measure-theoretical setting) non-perturbative localization results for the special case of the almost Mathieu potential. Shortly afterwards, J.Bourgain and his collaborators developed a non-perturbative approach that allows to treat one-dimensional quasiperiodic operators with arbitrary real analytic potentials. This approach combines several ideas from complex analysis, harmonic analysis, Diophantine approximation and semi-algebraic geometry. Modifications of this approach can also be applied to multi-dimensional models.

In this mini-course, we will consider the simplest non-trivial case of a quasiperiodic operator with arbitrary real analytic potential (one-dimensional and one-frequency), and establish non-perturbative Anderson localization for this operator using Bourgain’s method.

Our goal is to be as self-contained as reasonably possible. Spectral theorem for bounded self-adjoint operators and Schnol’s theorem will be stated without proofs. Knowledge of basic complex analysis and basic Fourier analysis is expected. No knowledge of semi-algebraic geometry will be assumed, but several facts will be stated without proofs.

Lecture notes and literature are available here: Notes and Literature.

### 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. Video recordings are available for download on Google Drive.

**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.