Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

2015/05/19 by Andrew Hassell, Hassell, Andrew, Victor Ivrii +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Dynamics and Fractals #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1505.04894

openalex publication_date 2015/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a compact Riemannian manifold with smooth boundary, and let R(λ) be the Dirichlet-to-Neumann operator at frequency λ. We obtain a leading asymptotic for the spectral counting function for λ^-1R(λ) in an interval [a₁, a₂) as λ→ ∞, under the assumption that the measure of periodic billiards on T^*M is zero. The asymptotic takes the form N(λ; a₁,a₂) = (κ(a₂)-κ(a₁))vol'(∂ M) λ^d-1+o(λ^d-1), where κ(a) is given explicitly by κ(a) = \fracω_d-1(2π)^d-1 \biggl( -(1)/(2π) ∫_-1¹ (1 - η²)^(d-1)/2 (a)/(a² + η²) dη- (1)/(4) + H(a) (1+a²)^(d-1)/2 \biggr) with the Heavyside function H(a).

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