Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q) with complex-valued periodic potential q when q is an element of L-1[0,1] and q(n) = 0 for n = 0, -1, -2, ..., where qn are the Fourier coefficients of q with respect to the system{e(iota 2 pi nx)}. We prove that the Bloch eigenvalues are (2 pi n + t)(2) for n is an element of Z, t is an element of C and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator. (C) 2014 Elsevier Inc. All rights reserved.