On geometric perturbations of critical Schr\"odinger operators with a surface interaction

TL;DR

This paper investigates how small geometric changes to a surface affect the spectral properties of critical Schrödinger operators with surface interactions, revealing conditions under which eigenvalues emerge or disappear.

Contribution

It provides a detailed analysis of spectral stability under surface deformations and introduces inequalities related to capacities of the surfaces.

Findings

Small radial deformations on a sphere induce eigenvalues.

General deformations can keep the spectrum empty.

Derived inequalities connect surface capacities and spectral properties.

Abstract

We study singular Schrodinger operators with an attractive interaction supported by a closed smooth surface A in R^3 and analyze their behavior in the vicinity of the critical situation where such an operator has empty discrete spectrum and a threshold resonance. In particular, we show that if A is a sphere and the critical coupling is constant over it, any sufficiently small smooth area preserving radial deformation gives rise to isolated eigenvalues. On the other hand, the discrete spectrum may be empty for general deformations. We also derive a related inequality for capacities associated with such surfaces.