On recovering non-local perturbation of non-self-adjoint Sturm - Liouville operator

Recently, there appeared a significant interest in inverse spectral problems for non-local operators arising in numerous applications. In the present work, we consider the operator with frozen argument ly = - y '' ( x ) + p(x)y(x) + q(x)y(a), which is a non-local perturbation of the non-self-adjoint Sturm - Liouville operator. We study the inverse problem of recovering the potential q is an element of L 2 (0, 7r ) by the spectrum when the coefficient p is an element of L 2 (0, 7r) is known. While the previous works were focused only on the case p = 0 , here we investigate the more difficult non-self-adjoint case, which requires consideration of eigenvalues multiplicities. We develop an approach based on the relation between the characteristic function and the coefficients {fin}n >= 1 of the potential q by a certain basis. We obtain necessary and sufficient conditions on the spectrum being asymptotic formulae of a special form. They yield that a part of the spectrum does not depend on q , i.e. it is uninformative. For the unique solvability of the inverse problem, one should supplement the spectrum with a part of the coefficients fi n , being the minimal additional data. For the inverse problem by the spectrum and the additional data, we obtain a uniqueness theorem and an algorithm.