### Welcome to the official site of the department of Mathematics and mathematical physics, Physics Faculty, Saint-Petersburg State University.

The site is under construction and filling-up. If you have any comments concerning functioning or content of the site, please, leave your comments under this record.

#### Everything you wanted to know, but were afraid to ask at a lecture

We have  an account at ask.fm, where you can ask questions. The answers for the best questions will be published on this site.

# Семинар 21 апреля

Заседание семинара кафедры высшей математики и математической физики состоится 21 апреля (среда) в 18-30.

Докладчик: Dmitri Vassiliev (University College London)

Тема: Invariant subspaces of elliptic systems

Абстракт:
Consider an elliptic self-adjoint pseudodifferential operator $$A$$ acting on m-columns of half-densities on a closed manifold $$M$$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of m orthonormal pseudodifferential projections commuting with the operator $$A$$ and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of $$L^2(M)$$ into invariant subspaces under the action of $$A$$ modulo $$C^\infty(M)$$. Furthermore, they allow us to decompose $$A$$ into m distinct sign definite pseudodifferential operators.

We use our pseudodifferential projections to show that the spectrum of $$A$$ decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of $$A$$. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems.

This is joint work with Matteo Capoferri (Cardiff). The talk is based on two of our papers, arXiv:2103.14325 and arXiv:2103.14334.

Как всегда Вы можете попасть на семинар, используя следующие данные:

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар 14 апреля

Заседание семинара кафедры высшей математики и математической физики состоится
14 апреля (среда) в 18-30.

Докладчик: Uzy Smilansky (Weizmann Institute of Science)

Тема: Multi-mode quantum graphs – a new approach (and results) of old and new problems

Абстракт:
We return to an old problem – the Schrӧdinger operator $H(x,y) = -\frac{\partial^2 \ }{\partial x^2} +\frac{1}{2} \left (-\frac{\partial^2 } {\partial y^2} + y^2 \right )+\lambda y \delta(x) \ ; \ \ \ x,y\in {\mathbf R^2}$ – a paradigm model for multi-mode quantum graphs, and discuss it using a scattering approach. This will enable shedding some light on previous results, and study some other related operators, e.g., the period case – the analogous Kronig Penney model.

Как всегда Вы можете попасть на семинар, используя следующие данные:

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар 7 апреля

Заседание семинара кафедры высшей математики и математической физики состоится
7 апреля (среда) в 18-30.

Докладчик: Karol Kozlowski (Univ Lyon)

Тема: Convergence of the form factor series in the Sinh-Gordon quantum field theory in 1+1 dimensions

Абстракт:
Within the approach of the bootstrap program, the physically pertinent observables in a massive integrable quantum field theory in 1+1 dimensions are expressed by means of the so-called form factor series expansion. This corresponds to a series of multiple integrals in which the nth summand is given by a n-fold integral. While being formally effective for various physical applications, so far, the question of convergence of such form factor series expansions was essentially left open. Still, convergence results are necessary so as to reach the mathematical well-definiteness of such construction and appear as necessary ingredients for the justification of numerous handlings that are carried out on such series and which play the role in the analysis of the physics at the roots of such models.

In this talks, I will first go through the physical origin as well as the various motivations for studying the convergence of form factor series expansions in massive quantum integrable field theories. Then, I will provide a description of the problem per se, in particular by discussing the Sinh-Gordon quantum field theory in 1+1 dimensions within the bootstrap program approach. Once this is settled, if time permits, I will discuss the the main features of the technique that allows one to prove this convergence. The proof amounts to obtaining a sufficiently sharp estimate on the leading large-n behaviour of the n-fold integral arising in this context. This appeared possible by refining some of the techniques that were fruitful in the analysis of the large-n behaviour of integrals over the spectrum of n\times n random Hermitian matrices.

Как всегда Вы можете попасть на семинар, используя следующие данные:

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар 31 марта

Заседание семинара кафедры высшей математики и математической физики состоится 31 марта (среда) в 18-30.

Докладчик: Jeffrey Galkowski (University College London)

Тема: Geodesic beams and Weyl remainders

Абстракт:
In this talk we discuss quantitative improvements for Weyl remainders under dynamical assumptions on the geodesic flow. We consider a variety of Weyl type remainders including asymptotics for the eigenvalue counting function as well as for the on and off diagonal spectral projector. These improvements are obtained by combining the geodesic beam approach to understanding eigenfunction concentration together with an appropriate decomposition of the spectral projector into quasimodes for the Laplacian. One striking consequence of these estimates is a quantitatively improved Weyl remainder on all product manifolds. This is joint work with Y.Canzani.

Как всегда Вы можете попасть на семинар, используя следующие данные:

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар кафедры

Семинар обычно проходит по средам в 18:30 онлайн в конференции Zoom
https://us02web.zoom.us/j/2709505573?pwd=dGZtbU9OaWVuWnVOVkk1Tm9kVXlrdz09

Meeting ID: 270 950 5573
Passcode: 779950

Теперь информация о семинаре кафедры ВММФ есть на страничке международного математического центра им. Эйлера

https://eimi.ru/math-physics-seminar/

Наш семинар включен в программу ММЦ им. Эйлера «Spectral Theory and Mathematical Physics».

# Семинар 24 марта

Заседание семинара кафедры высшей математики и математической физики состоится 24 марта (среда) в 18-30.

Докладчик: Andrew Comech, Texas A&M University, College Station, TX and IITP, Moscow

Тема: Virtual levels and virtual states of operators in Banach spaces

Абстракт:
Virtual levels admit several equivalent characterizations:
(1) there are corresponding eigenstates from L2 or a space “slightly weaker” than L2;
(2) there is no limiting absorption principle in the vicinity of a virtual level (e.g. no weights such that the “sandwiched” resolvent remains uniformly bounded);
(3) an arbitrarily small perturbation can produce an eigenvalue.
We develop a general approach to virtual levels in Banach spaces and provide applications to Schroedinger operators with nonselfadjoint potentials and in any dimension.

This is a joint work with Nabile Boussaid based on the preprint arXiv:2101.11979 [math.AP].

Как всегда Вы можете попасть на семинар, используя следующие данные:

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар 17 марта

Заседание семинара кафедры высшей математики и математической физики состоится 17 марта (среда) в 18-30.

Докладчик: Александр Соболев

Тема: Об абсолютной непрерывности операторов Теплица

Абстракт:
It has been known since the 1960’s that the spectra of Toeplitz operators are purely absolutely continuous. The aim of the talk is to give a short survey of relevant results and to describe a new spectral classification for Toeplitz operators with piecewise continuous symbols. The classification result is obtained by analysing scattering properties of Toeplitz operators.

The talk is based on a joint work with D.Yafaev (Rennes).

Как всегда Вы можете попасть на семинар, используя следующие данные:

Join Zoom Meeting
https://us02web.zoom.us/j/2709505573?pwd=dGZtbU9OaWVuWnVOVkk1Tm9kVXlrdz09

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар 10 марта

Заседание семинара кафедры высшей математики и математической физики состоится 10 марта (среда) в 18-30.

Докладчик: Alexander Pushnitski, King’s College London

Тема: Szegő-type limit theorems for “multiplicative Toeplitz” operators and non-Følner approximations

Абстракт:
We will discuss an analogue of the First Szegő Limit Theorem for multiplicative Toeplitz operators and highlight the role of the multliplicative Følner condition in this topic.

The talk is based on a joint work with Nikolai Nikolski.

Как всегда Вы можете попасть на семинар, используя следующие данные:

Join Zoom Meeting
https://us02web.zoom.us/j/2709505573?pwd=dGZtbU9OaWVuWnVOVkk1Tm9kVXlrdz09

Meeting ID: 270 950 5573
Passcode: 779950

# Семинар 3 марта

Заседание семинара кафедры высшей математики и математической физики состоится 3 марта (среда) в 18-30.

Докладчик: Григорий Владимирович Розенблюм

Тема: The Birman–Schwinger type operator with singular measure. Eigenvalues analysis, Connes integral and rectifiable sets

Абстракт:
This is an extended version of the authors’ talk on 28.12.2020 at the V.I.Smirnov seminar, containing some new results. We consider the Birman–Schwinger type operator $$\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}*P \mathfrak{A}$$, where P is a signed measure in $$\mathbb{R}^\mathbf{N}$$ and $$\mathfrak{A}$$ is a pseudodifferential operator in $$\mathbb{R}^\mathbf{N}$$ of order $$-l=-\mathbf{N}$$ (in the leading case, $$\mathfrak{A}= (1-\Delta)^{-\mathbf{N}/4}$$). Under rather general conditions we find eigenvalue estimates for this operator, and for measures supported on a Lipschitz surface, find eigenvalue asymptotics. The interesting case is when measure $$P$$ contains a singular component. A peculiar feature here is that the order of the eigenvalue estimates and asymptotics does not depend on the dimensional characteristics of the support of the measure, so, contributions of components of different dimensions just add up. Further on the results are carried over to more general measures supported on so-called rectifiable sets. We will discuss relation of our results to spectral theory of fractals, logarithmic potential and, finally, to noncommutative integration of singular measures.

Как всегда Вы можете попасть на семинар, используя следующие данные:

Meeting ID: 270 950 5573
Passcode: 779950