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

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 $${\mathbf{H}}_{R,\alpha }=-\mathrm{\Delta }+\sum _{k=1}^{\mathrm{\infty }}{\alpha }_k\delta (\mathrm{\mid }x\mathrm{\mid }-r_k)\mathrm{.}$$ 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 ${\mathbf{H}}_{R,\alpha }$.