- journal article metadata
European Mathematical Society Publishing House
2017-10-01 23:40:02
Journal of Spectral Theory
J. Spectr. Theory
JST
1664-039X
1664-0403
Quantum theory
10.4171/JST
http://www.ems-ph.org/doi/10.4171/JST
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2017
3
Spectral asymptotics for the semiclassical Dirichlet to Neumann operator
Andrew
Hassell
Australian National University, Canberra, Australia
Victor
Ivrii
University of Toronto, Canada
Dirichlet-to-Neumann operator, semiclassical Dirichlet-to-Neumann operator, spectral asymptotics
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 \begin{equation*} \mathrm N(\lambda; a_1,a_2) = ( \kappa(a_2)-\kappa(a_1))\mathrm {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}} \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) .\end{equation*}
Partial differential equations
Global analysis, analysis on manifolds
881
905
10.4171/JST/180
http://www.ems-ph.org/doi/10.4171/JST/180