Characterization of the spectrum of Schrödinger operators with periodic distributions

Abstract

Consider the Schrödinger operator Ly = −y″+q′y on L2(ℝ) with a distribution q′, where q ∈ L2(0,1) is a 1-periodic function. The spectrum of L is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) both in terms of vertical slits on the quasimomentum domain and in terms of gap lengths. Furthermore, we obtain a priori two-sided estimates for these maps. In addition, we solve the inverse problem and obtain the two-sided estimates for the Riccati map. Using these results we determine various asymptotics for our operator: asymptotics of the gap lengths, of periodic spectrum, and of the Dirichlet eigenvalues.