Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

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

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