The Spectral Expansion for a Nonself-adjoint Hill Operator with a Locally Integrable Potential

We construct the spectral expansion for the one-dimensional Schr¨odinger operator L = − d2 dx2 + q x −∞ < x < ∞ in L2 −∞ ∞ , where q x is a 1-periodic, Lebesgue integrable on [0,1], and complex-valued potential. We obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator Lt , for t = 0, π, generated by this operation in L2 0 1 and the t-periodic boundary conditions. Using it, we prove that the eigenfunctions and associated functions of Lt form a Riesz basis in L2 0 1 for t = 0, π. Then we find the spectral expansion for the operator L.