Schrödinger operators with concentric $δ$-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 $$ \mathbf{H}_{R,α}=-Δ+\sum_{k=1}^\inftyα_kδ(|x|-r_k). $$ We obtain necessary and sufficient conditions for the operator $\mathbf{H}_{R,α}$ to be self-adjoint, lower-semibounded. Also we investigate the spectral types of $\mathbf{H}_{R,α}$.

24 pages