Seminar in Mathematical Physics

In 2021, this seminar is a part of the EIMI thematic program “Spectral Theory and Mathematical Physics“. The meetings are held weekly on Wednesdays at 18:30 Moscow Time (15:30 UTC) unless otherwise noted. To subscribe to our mailing list and receive zoom meeting information, please sign up online.

The seminar is organized by the Department of Mathematics and Mathematical Physics at SPbU within the program “Spectral Theory and Mathematical Physics“. If you have any questions, please contact Tatiana Suslina and Nikita Senik.

Upcoming talks

March 3, 18:30 MSK

Grigori Rozenblum, Chalmers University of Technology

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

 

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.

March 10, 18:30 MSK

Alexander Pushnitski, King’s College London

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

 

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.

 

March 17, 18:30 MSK

Alexander Sobolev, University College London

TBA

 

 

March 24, 18:30 MSK

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

Virtual levels and virtual states of operators in Banach spaces

 

Virtual levels admit several equivalent characterizations:
(1) there are corresponding eigenstates from L^2 or a space “slightly weaker” than L^2;
(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].

 

Past talks

February 24, 19:10 MSK

Valery Smyshlyaev, University College London

Whispering gallery waves near boundary inflection: a canonical high-frequency diffraction problem

 

A particular problem of interest is that of a whispering gallery high-frequency asymptotic mode propagating along a concave part of a boundary and then scattering at a boundary inflection point. Like Airy ODE and associated Airy function are fundamental for describing transition from oscillatory to exponentially decaying asymptotic behaviors, the boundary inflection problem leads to an arguably equally fundamental canonical boundary-value problem for a special PDE describing transition from a “modal” to a “scattered” high-frequency asymptotic behaviors. This is a Schrödinger equation on a half-line with a potential linear in both space and time. The latter problem was first formulated and analysed by M.M.Popov starting from 1970-s. The associated solutions have asymptotic behaviors with a discrete spectrum at one end and with a continuous spectrum at the other end, and of central interest is to find the map connecting the above two asymptotic regimes. The problem however lacks separation of variables, except in the asymptotical sense at both of the above ends.

Nevertheless, we perform a non-standard perturbation analysis at the continuous spectrum end, ultimately describing the desired map connecting the two asymptotic representations. We also show that the problem permits a re-formulation as a one-dimensional boundary integral equation, whose regularization allows its further asymptotic analysis.

February 17, 18:30 MSK

Sergey Yu. Dobrokhotov, Ishlinsky Institute for Problems in Mechanics RAS, Moscow

Real-valued semiclassical approximation for the asymptotics with complex-valued phases for Plancherel-Rotach asymptotics of Hermitian type orthogonal polynomials

 

The Hermitian type orthogonal polynomials H_{n_1,n_2}(z,a) are determined by the pair of recurrence relations for the polynomials

    \[H_{n_1+1,n_2}(z,a), H_{n_1,n_2+1}(z,a), H_{n_1,n_2-1}(z,a), H_{n_1,n_2}(z,a).\]

We obtain a uniform asymptotics of diagonal polynomials H_{n,n}(z,a) in the form of an Airy function for n>>1, which is a far-reaching generalization of the Plancherel-Rotach asymptotic formulas for Hermitian polynomials. We discuss one of the possible approach which we call “real-valued semiclassics for asymptotics with complex-valued phases” (another approach based on the construction of decompositions of bases of homogeneous difference equations was recently developed by A.I.Aptekarev and D.N.Tulyakov). This approach can be applied to construct asymptotic formulas for various orthogonal polynomials. Introducing an artificial small parameter h=O(1/n) and a continuous function \varphi(x,z,a) such that H(z,a)=\varphi(kh,z,a), we reduce the described to a pseudo-differential equation for \varphi, where x is a variable and (z,a) are parameters. Seeking its solution in the WKB-form, one obtains the Hamilton-Jacobi equations with complex Hamiltonians connected with a third-order algebraic curve. This circumstance is the main difficulty of solving the problem and, as a rule, leads to the transition from the real variable x to the complex one. In this problem, we propose a different approach based on a reduction of the original problem to three equations, two of which have asymptotics with a purely imaginary phase, and the symbol of the third one is pure real and has the form \cos ⁡p+V_0 (x)+hV_1 (x)+O(h^2). This ultimately allows us to represent the desired asymptotic uniformly through the Airy function of the complex but real-valued argument.

The talk is based on a joint work with A.I.Aptekarev (Keldysh Institute of Applied Mathematics RAS), D.N.Tulyakov (Keldysh Institute of Applied Mathematics RAS) and A.V.Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS).