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 $$\mathbf{H}_{R,\alpha}=-\Delta+\sum_{k=1}^\infty\alpha_k\delta(|x|-r_k).$$ 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 $\mathbf{H}_{R,\alpha}$.