Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

TL;DR

This paper derives the leading asymptotic behavior of the spectral counting function for the semiclassical Dirichlet-to-Neumann operator on a compact Riemannian manifold with boundary, under certain dynamical assumptions.

Contribution

It provides a precise asymptotic formula for the spectral counting function of the Dirichlet-to-Neumann operator in the semiclassical limit, assuming zero measure of periodic billiards.

Findings

Asymptotic formula for spectral counting function derived

Explicit expression for the function ppa(a) provided

Results depend on the measure-zero condition for periodic billiards

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 $H(a)$.