Spectral Asymptotics for the Semiclassical Dirichlet to Neumann Operator

Let M be a compact Riemannian manifold with smooth boundary, and let $$R(\lambda )$$ be the Dirichlet-to-Neumann operator at frequency $$\lambda $$ . The semiclassical Dirichlet-to-Neumann operator $$R_{\text{scl}}(\lambda) $$ is defined to be $$\lambda^{-1} R(\lambda) $$ . We obtain a leading asymptotic for the spectral counting function for $$R_{\text{scl}}(\lambda) $$ 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 $$\mathsf{N}(\lambda; a_1,a_2) = \bigl( \kappa(a_2)-\kappa(a_1)\bigr)\text{vol}'(\partial M) \lambda^{d-1}+o(\lambda^{d-1}), $$ where $$\kappa(a) $$ is given explicitly by $$\begin{aligned}\kappa(a) &= \frac{\omega_{d-1}}{(2\pi)^{d-1}} \bigg( -\frac{1}{2\pi} \int_{-1}^1 (1 - \eta^2)^{(d-1)/2} \frac{a}{a^2 + \eta^2} \, d\eta \\& \qquad\qquad\qquad\qquad - \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \bigg) .\end{aligned} $$