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

In this paper, we give a description of the spectral analysis of the Schrodinger operator L(q) with the potential q(x) = 4cos(2)x + 4iVsin2x 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 V-2 < V-3 < such that the operator L(q) has no ESS and has ESS, respectively, if and only if V not equal V-k and V = V-k for k >= 2, where V-k -> infinity as k -> infinity, V-2 is the second critical point. Using ESS, we classify the spectral expansion in term of the points V-k for k >= 2. Finally, we discuss the critical points V-k, formulate some conjectures, and describe the changes in the spectrum of L(q) when V changes from 1/2 to infinity.