*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.