В рамках семинара кафедры высшей математики и математической физики пройдет 13 и 20 октября (среда)  в 18-00 пройдет миникурс лекций.

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

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

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Meeting ID: 270 950 5573          Passcode: 779950

Абстракт:
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.

Планируется, что материал будет излагаться просто и для неспециалистов.

Заседание семинара кафедры высшей математики и математической физики состоится
6 октября (среда) в 18-00.

Докладчик: А.А. Федотов

Тема: О спектре несамосопряженного квазипериодического оператора
(по совместной работе с Д.И. Борисовым)

Абстракт:
Будет рассказано о спектральных свойствах простейшего нетривиального несамосопряженного одномерного квазипериодического разностного оператора Шредингера, о котором можно рассказывать студентам на лекциях.

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:

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Meeting ID: 270 950 5573
Passcode: 779950

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.

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

Meeting ID: 270 950 5573
Passcode: 779950

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:

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

Meeting ID: 270 950 5573
Passcode: 779950