Aftermath

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:

My talk at PDE & Analysis Seminar (Hebrew University, December 6, 2017)

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

My talk at Mathematical Analysis and Applications Seminar (Weizmann, December 5, 2017)

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 talk at PDEs and Analysis Seminar (ANU, November 7, 2017)

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

My talk at joint UCLA/Caltech Analysis seminar (October 20, 2017)

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

My talk at Claremont Graduate University Colloquium (October 18, 2017)

Etudes of Spectral Theory

I briefly describe five old but still actively explored problems of the Spectral Theory of Partial Differential Equations

1.  How eigenvalues are distributed (where eigenvalues often mean squares of the frequencies in the mechanical or electromagnetic problems or energy levels in the quantum mechanics models) and the relation to the behaviour of the billiard trajectories.
2.  Equidistribution  of eigenfunctions and connection to ergodicity of billiard trajectories (a quantum chaos and  a classical chaos).

3.  Can one hear the shape of the drum?
4.  Nodal lines and Chladni plates.
5.  Strange spectra of quantum systems.

Slides

Dmitry Jakobson made several very useful remarks and suggestions