Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer

TL;DR

This paper analyzes the spectral properties of the Dirichlet Laplacian on a generalized parabolic layer, revealing the asymptotic distribution of its discrete eigenvalues through reduction to an effective one-dimensional Schrödinger operator.

Contribution

It provides the first detailed asymptotic analysis of the discrete spectrum for the Dirichlet Laplacian on unbounded parabolic layers, linking geometry to spectral asymptotics.

Findings

Discrete spectrum asymptotics derived

Reduction to one-dimensional Schrödinger operator

Spectral behavior determined by layer geometry at infinity

Abstract

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $\mathsf{H}$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $\mathsf{H}$ has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for $\mathsf{H}$ by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schr\"odinger 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 $\mathcal{P}$ at infinity.