On periodic continuation of soliton potentials beyond abounded interval

Abstract

The well-known soliton potentials, as defined on a bounded interval of the realline and continued beyond by periodicity, are considered. In such a way oneobtains new exactly solvable periodic potentials for the Schrödingerequation. It is found that the energy band edges are determined by atranscendental equation which is very similar to the corresponding equationfor the Kronig-Penney potential. Solutions of the Schrödinger equation areexpressed in terms of elementary functions. The potentials involved mightbecome non-standard textbook examples with possible applications insolid-state physics.