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 $q\in L_2(0, \pi)$ by the spectrum when the coefficient $p\in L_2(0, \pi)$ 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 $\{ \xi_n\}_{n \ge 1}$ 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 $ \xi_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.