On the Bound States of Schr\"odinger Operators with $\delta$-interactions on Conical Surfaces

TL;DR

This paper analyzes the spectral properties of Schrödinger operators with delta interactions on conical surfaces in dimensions three and higher, revealing dimension-specific discrete spectra and their dependence on cone geometry.

Contribution

It demonstrates that discrete spectra occur only in three dimensions, are axisymmetric, and depend monotonically on the cone's aperture, with detailed spectral accumulation behavior.

Findings

Discrete spectrum exists only in three dimensions.

Eigenvalues are axisymmetric and increase with cone aperture.

Spectral accumulation follows a logarithmic pattern.

Abstract

In dimension greater than or equal to three, we investigate the spectrum of a Schr{\"o}dinger operator with a $\delta$-interaction supported on a cone whose cross section is the sphere of co-dimension two. After decomposing into fibers, we prove that there is discrete spectrum only in dimension three and that it is generated by the axisymmetric fiber. We get that these eigenvalues are non-decreasing functions of the aperture of the cone and we exhibit the precise logarithmic accumulation of the discrete spectrum below the threshold of the essential spectrum.