Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Andrew Hassell; Victor Ivrii

doi:10.4171/jst/180

JournalsjstVol. 7, No. 3pp. 881–905

Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Andrew Hassell
Australian National University, Canberra, Australia
Victor Ivrii
University of Toronto, Canada

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Abstract

Let $M$ be a compact Riemannian manifold with smooth boundary, and let $R (λ)$ be the Dirichlet–to–Neumann operator at frequency $λ$ . The semiclassical Dirichlet–to–Neumann operator $R_{scl} (λ)$ is defined to be $λ^{- 1} R (λ)$ . We obtain a leading asymptotic for the spectral counting function for $R_{scl} (λ)$ 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

N (λ; a_{1}, a_{2}) = (κ (a_{2}) - κ (a_{1})) vol^{'} (\partial M) λ^{d - 1} + o (λ^{d - 1}),

where $κ (a)$ is given explicitly by

κ (a) = \frac{ω _{d - 1}}{( 2 π ) ^{d - 1}} (- \frac{1}{2 π} \int_{- 1}^{1} (1 - η^{2})^{(d - 1) /2} \frac{a}{a ^{2} + η ^{2}} d η - \frac{1}{4} + H (a) (1 + a^{2})^{(d - 1) /2}) .

Cite this article

Andrew Hassell, Victor Ivrii, Spectral asymptotics for the semiclassical Dirichlet to Neumann operator. J. Spectr. Theory 7 (2017), no. 3, pp. 881–905

DOI 10.4171/JST/180