Spectrum of Dirichlet Laplacian in a conical layer

TL;DR

This paper investigates the spectral properties of the Dirichlet Laplacian in a conical layer, revealing an infinite sequence of bound states below the continuum threshold and providing numerical examples of eigenfunctions.

Contribution

It provides a detailed analysis of the eigenvalues and eigenfunctions of the Dirichlet Laplacian in a conical layer, including numerical computations of the bound states.

Findings

Existence of infinitely many eigenvalues below the continuum threshold

Characterization of geometrically induced bound states

Numerical examples of eigenfunctions

Abstract

We study spectral properties of Dirichlet Laplacian on the conical layer of the opening angle $\pi-2\theta$ and thickness equal to $\pi$. We demonstrate that below the continuum threshold which is equal to one there is an infinite sequence of isolated eigenvalues and analyze properties of these geometrically induced bound states. By numerical computation we find examples of the eigenfunctions.