Victor Ivrii' Rumblings and Musings

Conference on Partial Differential Equations and Applications
in Memory of Professor B.Yu. Sternin

November 6–9, 2018, Moscow, Russia

10th St. Petersburg Conference in Spectral Theory
dedicated to the memory of M. Sh. Birman

9 – 12 June, 2018 Euler Institute, St. Petersburg, Russia

Microlocal Analysis, Sharp Spectral Asymptotics and Applications

Finally!

Spectral asymptotic for Steklov’s problem in domains with edges (work in progress)

We derive sharp eigenvalue asymptotics for Dirichlet-to-Neumann operator in the domain with edges and discuss obstacles for deriving a sharper (two-term) asymptotics

Slides

My Research Monograph is almost ready (in January I plan to edit Introduction, Preface and add some references, after which I pronounce it “Done” and I am not going to add any new material, at least for several years).

Meanwhile I started several new topics:

• Asymptotics of the ground state energy for relativistic heavy atoms and molecules, see Paper at arXiv: 1708.07737 and Presentation.
• Spectral asymptotic for Steklov’s problem in domains with edges, the paper is in progress, see Presentation.

Asymptotics of the ground state energy for relativistic heavy atoms and molecules

We discuss sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, without magnetic field or with the self-generated magnetic field, and, in particular, relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. In particular, we conclude that the Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.

Slides

Spectral asymptotic for Steklov’s problem in domains with edges (work in progress)

We derive sharp eigenvalue asymptotics for Dirichlet-to-Neumann operator in the domain with edges and discuss obstacle for deriving a sharper (two-term) asymptotics

Slides

Spectral asymptotic for Steklov’s problem in domains with edges (work in progress)

We derive sharp eigenvalue asymptotics for Dirichlet-to-Neumann operator in the domain with edges and discuss obstacles for deriving a sharper (two-term) asymptotics

Slides

Spectral asymptotics for fractional Laplacians

Consider a compact domain with the smooth boundary in the Euclidean space. Fractional Laplacian is defined on functions supported in this domain as a (non-integer) power of the positive Laplacian on the whole space restricted then to this domain. Such operators appear in the theory of stochastic processes.

It turns out that the standard results about distribution of eigenvalues (including two-term asymptotics) remain true for fractional Laplacians. There are however some unsolved problems.

Slides

Eigenvalue Asymptotics for Dirichlet-to-Neumann Operator

Let $X$  be a compact manifold with the boundary $Y$ and   $R(k)$  be a Dirichlet-to-Neumann operator: $R (k):f \to  \partial_n u |_Y$ where u  solves
$$
(\Delta+k^2) u=0,  \  u|_Y=f.
$$
We establish asymptotics as $k\to \infty$  of the number of eigenvalues of   $k^{-1}R (k)$ between $a$  and  $b$.

We will discuss tools, used to solve this problem: sharp semiclassical spectral asymptotics and Birman-Schwinger principle.

This is a joint work with Andrew Hassell, Australian National University.

Slides