Discrete spectrum of interactions concentrated near conical surfaces

TL;DR

This paper investigates the spectral properties of operators related to conical geometries, revealing infinite eigenvalues accumulating below the essential spectrum threshold, with accumulation rates linked to auxiliary one-dimensional operators.

Contribution

It provides a detailed analysis of the discrete spectrum near conical surfaces for specific operators, including explicit accumulation rate formulas.

Findings

Operators have infinitely many eigenvalues below the essential spectrum.

Eigenvalues accumulate at a rate determined by a 1D curvature-induced potential.

Results apply to conical layers and delta-interaction Schrödinger operators.

Abstract

We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schr\"odinger operators with attractive $\delta$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.