A new treatment for some periodic Schrödinger operators

Wei He

Published 2014-12-21Version 1

We revise some aspects of the asymptotic solution for the eigenvalues for Schr\"odinger operators with periodic potential, from the perspective of the Floquet theory. In the context of classical Floquet theory, when the periodic potential can be treated as small perturbation we give a new method to compute the asymptotic spectrum. For elliptic potentials a generalized Floquet theory is needed. In order to produce other asymptotic solutions consistent with known results, new relations for the Floquet exponent and the monodromy of wave function are proposed. Many Schr\"odinger equations of this type, such as the Hill's equation and the ellipsoidal wave equation, etc., can be treated by this method.