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∈L1[0,1] and qn=0 for n=0,−1,−2,..., where qn are the Fourier coefficients of q with respect to the system {ei2πnx}. We prove that the Bloch eigenvalues are (2πn+t)2 for n∈Z, t∈C and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.