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.