Spectral analysis of the Schrödinger operator with a PT-symmetric periodic optical potential

Abstract

In this paper, we give a description of the spectral analysis of the Schrödinger operator L(q) with the potential q(x) = 4 cos2 x + 4iV sin 2x for all V > 1/2. First, we consider the Bloch eigenvalues and spectrum of L(q). Then, we investigate spectral singularities and essential spectral singularities (ESS). We prove that there exists a sequence V2 < V3 < ⋯ such that the operator L(q) has no ESS and has ESS, respectively, if and only if V ≠ Vk and V = Vk for k ≥ 2, where Vk → ∞ as k → ∞, V2 is the second critical point. Using ESS, we classify the spectral expansion in term of the points Vk for k ≥ 2. Finally, we discuss the critical points Vk, formulate some conjectures, and describe the changes in the spectrum of L(q) when V changes from 1/2 to ∞.