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Spectral Asymptotics for the Semiclassical Dirichlet to Neumann Operator

  • Andrew HassellEmail author
  • Victor Ivrii
Chapter

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 \(R_{\text{scl}}(\lambda) \) 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
$$\begin{aligned}\kappa(a) &= \frac{\omega_{d-1}}{(2\pi)^{d-1}} \bigg( -\frac{1}{2\pi} \int_{-1}^1 (1 - \eta^2)^{(d-1)/2} \frac{a}{a^2 + \eta^2} \, d\eta \\& \qquad\qquad\qquad\qquad - \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \bigg) .\end{aligned} $$

Key words and phrases

Dirichlet-to-Neumann operator semiclassical Dirichlet-to-Neumann operator spectral asymptotics 

2010 Mathematics Subject Classification:

35P20 58J50 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematical Sciences Institute, Australian National UniversityCanberraAustralia
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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