Schrödinger operators with concentric $\delta$-shells

Abstract

We investigate the spectral properties of the Schrödinger 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 ${\mathbf{H}}_{R,\alpha }$.

24 pages