A spectral isoperimetric inequality for cones

TL;DR

This paper establishes a spectral isoperimetric inequality for Schrödinger operators with delta interactions supported on smooth cones, identifying geometric minimizers for the principal eigenvalue using novel test function constructions.

Contribution

It introduces a Faber-Krahn-type inequality for these operators and presents a new method for constructing test functions in the Birman-Schwinger framework.

Findings

Circles are unique minimizers for the energy functional.

The principal eigenvalue is minimized by specific geometric configurations.

New test function construction technique for spectral inequalities.

Abstract

In this note we investigate three-dimensional Schr\"odinger operators with $\delta$-interactions supported on $C^2$-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.