Microlocal Analysis, Sharp Spectral Asymptotics and Applications
Finally!
Like it or hate it—I don't care
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
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
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.
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
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
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.
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.
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.
Dmitry Jakobson made several very useful remarks and suggestions