On the local solvability and stability of the partial inverse problems for the non-self-adjoint Sturm–Liouville operators with a discontinuity

In this work, we study the inverse spectral problems for the Sturm-Liouville operators on [0, 1] with complex coefficients and a discontinuity at x = a ∈ (0, 1). Assume that the potential on (a, 1) and some parameters in the discontinuity and boundary conditions are given. We recover the potential on (0, a) and the other parameters from the eigenvalues. This is the so-called partial inverse problem. The local solvability and stability of the partial inverse problems are obtained for a ∈ (0, 1), in which the error caused by the given partial potential is considered. Moreover, we obtain two new uniqueness theorems for the partial inverse problem. The results here generalize the previous work of Yang and Bondarenko [J. Differ. Equations 268, 6173–6188 (2020)], which only considered the self-adjoint case for a ∈ (0, 1/2] and did not take into account the error caused by the given partial potential.