Online Mini-Courses in
Spectral Theory and Mathematical Physics
3–26 November 2020
These are warm-up mini-courses for the EIMI thematic program on Spectral Theory and Mathematical Physics to be held in St. Petersburg in 2021 (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 on Zoom, and registration is required to obtain the link. Click here to register.
Also see the timetable page for the schedule.
Find the video recordings here.
Alexander Pushnitski (King's College London, UK)
Schmidt subspaces of Hankel operators
3–5 November, 2020
Let Γ be a compact Hankel operator acting on the Hardy class H² over the unit circle. The purpose of the lectures is to discuss the structure of the Schmidt spaces of Γ (i.e. the eigenspaces of Γ ⃰ Γ) as a class of subspaces of H².
It turns out that the Schmidt spaces of Γ are the images of model spaces under the action of isometric multipliers. The action of Γ on the Schmidt spaces can also be explicitly described. All of these notions will be introduced and discussed in detail in the lectures. If time permits, an inverse spectral problem for Γ will be briefly described.
The lectures are based on recent joint work of the author with Patrick Gérard (Orsay).
Lecture notes are available here, and video recordings, here.
Alexander Sobolev (University College London, UK)
Recent analytic and spectral results for the multi-particle Schrödinger operator
17–19 November 2020
The lectures focus on the properties of the one-particle density matrix γ(x, y), x, y ∈ ℝ³. This is one of the key objects in the quantum-mechanical approximation schemes. The aim is to present the following results obtained recently:
- the asymptotic formula for the eigenvalues of the self-adjoint operator Γ with the kernel γ(x, y),
- real analyticity of the function γ(x, y).
Lecture 1: Background and results (slides, video)
Lecture 2: Compact operators (slides, video)
Lecture 3: Spectrum of the one-particle density matrix (slides, video)
Lecture 4: Real analyticity of the density matrix (slides, video)
Alexander Its (IUPUI, USA & St. Petersburg University, Russia)
Toeplitz determinants and Painlevé transcendents. A Riemann–Hilbert point of view
2–3 December, 2020
Starting with Onsager's celebrated solution of the two-dimensional Ising model in the 1940's, Toeplitz and Hankel determinants have been one of the principal analytic tools in modern mathematical physics; specifically, in the theory of exactly solvable statistical mechanics, and quantum field models, and in the theory of random matrices.
The main analytical issue of the theory of Toeplitz determinants is their large size asymptotic behavior. There are two complementary approaches to study this question. The (historically) first approach is based on the general operator techniques, and it has been used in the theory of Toeplitz and Hankel determinants since the classical works of Szego and Widom. The second approach is younger, and it is based on the Riemann-Hilbert method of the theory of integrable systems.
In this mini course, the essence of the Riemann-Hilbert method in the theory of Topelitz determinants will be presented. The focus will be on the use of the method to obtain the Painlevé type description of the transition asymptotics of Toeplitz determinants. The Riemann-Hilbert view on the Painlevé functions will be also explained.
If time permits, some most recent results related to the so-called bordered Toeplitz determinants and Toeplitz + Hankel determinants will be discussed.
Slides are available here, and video recordings, here.