Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer

2018/05/31 by Exner, Pavel, Lotoreichik, Vladimir
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1805.12448

Abstract

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian H on an unbounded, radially symmetric (generalized) parabolic layer P⊂ℝ³. It was known before that H has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for H by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrödinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer P at infinity.

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