# 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