An upper bound on the allowed bands of the Bloch spectrum of one-dimensional Schrödinger operators with periodic potentials

Abstract

Using the principle of increasing flux and its complement, an upper bound associated with the allowed bands of the Bloch spectrum of one-dimensional Schrödinger operators with periodic potentials is derived. Namely, let a be the period of the potential, E0 the lower bound of the Bloch spectrum and (E2( j−1),E2j−1) the allowed bands ( j=1,2,...). Then π>a[(E2j−1−E0)1/2 −(E2(j−1)−E0)1/2].