On a Class of Non-self-adjoint Multidimensional Periodic Schrodinger Operators

We investigate the multidimensional Schrodinger operator L(q) with complex-valued periodic, with respect to a lattice, potential q when the Fourier coefficients of q with respect to the orthogonal system {exp(i(a,x))}, where a changes in the dual lattice, vanish if a belong to a half-space We prove that the Bloch eigenvalues of L(q) and of the free operator L(0) are the same and find explicit formulas for the Bloch functions. It implies that the Fermi surfaces of L(q) and L(0) are the same. The considered set of operators includes a large class of PT symmetric operators used in the PT symmetric quantum theory.