Spectral asymptotics for the Dirichlet-to-Neumann operator at high frequency

Abstract

Let $\Omega$ be a Euclidean domain with smooth boundary, and let $\lambda$ be a positive real number.

The Dirichlet-to-Neumann operator at frequency $\lambda$, denoted $R\left(\lambda \right)$, maps smooth functions on the boundary of $\Omega$ to smooth functions on the boundary of $\Omega$. It is defined as follows: if $f\in C^{\infty }\left(\partial \Omega \right)$, then we find the function $u$ defined in the interior of $\Omega$ satisfying the Helmholtz equation $\left(-\Delta -{\lambda }^2\right)u=0$, and with boundary value $f$. This can be solved uniquely provided that ${\lambda }^2$ is not a Dirichlet eigenvalue of $-\Delta$. Then $R\left(\lambda \right)f$ is defined to be the normal derivative of $u$ at $\partial \Omega$.

The operator $R\left(\lambda \right)$ is intimately related to eigenvalue counting functions for the domain $\Omega$. For example, the number of negative eigenvalues of $R\left(\lambda \right)$ is equal to the difference between the Neumann and Dirichlet eigenvalue counting functions (at eigenvalue ${\lambda }^2$).

In this talk, I will explain how to get a leading asymptotic for the number of eigenvalues of $R\left(\lambda \right)$ in the interval $\left[a\lambda ,b\lambda \right)$ as $\lambda \to \infty$, given a condition on the periodic billiard trajectories in $\Omega$.