Schr\"odinger operators with concentric $\delta$-shells

TL;DR

This paper studies the spectral properties of Schr"odinger operators with singular interactions supported on an infinite set of concentric spheres, providing conditions for self-adjointness, boundedness, and analyzing their spectral types.

Contribution

It offers necessary and sufficient conditions for the self-adjointness and lower-semiboundedness of Schr"odinger operators with concentric delta-shells, and investigates their spectral characteristics.

Findings

Established criteria for self-adjointness of the operators.

Determined conditions for lower-semiboundedness.

Analyzed the spectral types of the operators.

Abstract

We investigate the spectral properties of the Schr\"odinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$ \mathbf{H}_{R,\alpha}=-\Delta+\sum_{k=1}^\infty\alpha_k\delta(|x|-r_k). $$ We obtain necessary and sufficient conditions for the operator $\mathbf{H}_{R,\alpha}$ to be self-adjoint, lower-semibounded. Also we investigate the spectral types of $\mathbf{H}_{R,\alpha}$.