Discrete spectrum for n-cell potentials

Abstract

We study the scattering problem, the Sturm-Liouville problem and the spectral problem with periodic or skew-periodic boundary conditions for the one-dimensional Schr\"odinger equation with an $n$-cell (finite periodic) potential. We give explicit upper and lower bounds for the distribution functions of discrete spectrum for these problems. For the scattering problem we give, besides, explicit upper and lower bounds for the distribution function of discrete spectrum for the case of potential consisting of $n$ not necessarily identical cells. For the scattering problem some results about transmission resonances are obtained.