Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Abstract

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_{\mathrm {scl}}(\lambda) is defined to be \lambda^{-1} R(\lambda) . We obtain a leading asymptotic for the spectral counting function for R_{\mathrm {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 \mathrm N(\lambda; a_1,a_2) = ( \kappa(a_2)-\kappa(a_1))\mathrm {vol}'(\partial M) \lambda^{d-1}+o(\lambda^{d-1}), where \kappa(a) is given explicitly by \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 - \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \bigg) .

Locations

• Journal of Spectral Theory - View
• arXiv (Cornell University) - View - PDF