Spectral asymptotics for $\delta$-interactions on sharp cones

TL;DR

This paper analyzes the spectral properties of 3D Schrödinger operators with delta interactions on sharp cones, revealing asymptotic behaviors of eigenvalues as the cone aperture narrows, with results involving transcendental equations and Bessel functions.

Contribution

It provides new asymptotic descriptions of eigenvalues for delta-interactions on sharp cones, extending understanding of spectral behavior in geometric singularities.

Findings

Eigenvalues exhibit specific asymptotic behavior as cone sharpness increases.

Eigenvalue counting function's asymptotics are characterized.

Numerical constants are derived from solutions to transcendental equations involving Bessel functions.

Abstract

We investigate the spectrum of three-dimensional Schr\"{o}dinger operators with $\delta$-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the threshold of the essential spectrum. We focus on spectral properties for sharp cones, that is when the cone aperture goes to zero, and we describe the asymptotic behavior of the eigenvalues and of the eigenvalue counting function. A part of the results are given in terms of numerical constants appearing as solutions of transcendental equations involving modified Bessel functions.