# Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Research paper by **Andrew Hassell, Victor Ivrii**

Indexed on: **21 Jun '15**Published on: **21 Jun '15**Published in: **Mathematics - Spectral Theory**

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#### Abstract

Let $M$ be a compact Riemannian manifold with smooth boundary, and let
$R(\lambda)$ be the Dirichlet-to-Neumann operator at frequency $\lambda$. We
obtain a leading asymptotic for the spectral counting function for
$\lambda^{-1}R(\lambda)$ in an interval $[a_1, a_2)$ as $\lambda \to \infty$,
under the assumption that the measure of periodic billiards on $T^*M$ is zero.
The asymptotic takes the form \begin{equation*} N(\lambda; a_1,a_2) =
\bigl(\kappa(a_2)-\kappa(a_1)\bigr)\mathsf{vol}'(\partial M)
\lambda^{d-1}+o(\lambda^{d-1}), \end{equation*} where $\kappa(a)$ is given
explicitly by \begin{equation*} \kappa(a) = \frac{\omega_{d-1}}{(2\pi)^{d-1}}
\biggl( -\frac{1}{2\pi} \int_{-1}^1 (1 - \eta^2)^{(d-1)/2} \frac{a}{a^2 +
\eta^2} \, d\eta - \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \biggr) \end{equation*}
with the Heavyside function $H(a)$.