Complete solution of the second inverse spectral problem (recover q from known M and two spectra) for the Sturm–Liouville operator with convolution perturbation

Establish a complete solution of the inverse spectral problem for the convolution-perturbed Sturm–Liouville operator L(q, M) := −y'' + q(x) y + ∫_0^x M(x−t) y(t) dt on (0, π), in which the convolution kernel M(x) is known and the Dirichlet and Dirichlet–Neumann spectra {λ_k} and {μ_k} are given; specifically, determine the potential q(x) ∈ L2(0, π) that produces these spectra and fully resolve the problem of reconstructing q from M and the two spectra.

Background

The paper studies the Sturm–Liouville operator perturbed by an integral operator with a convolution kernel, L(q, M) = −y'' + q(x) y + ∫_0^x M(x−t) y(t) dt, with q ∈ L2(0, π) and M(x) = (π−x)^{-1} M_0(x), M_0 ∈ L2(0, π). Two inverse problem formulations are discussed historically. The first, reconstructing M from the Dirichlet spectrum, has received substantial progress and complete solutions in prior work.

The second inverse problem asks to reconstruct q(x) when M is known and both Dirichlet and Dirichlet–Neumann spectra are given. Although there have been further investigations, the authors explicitly state that the methods leading to a complete solution are not clear, indicating that the problem remains unresolved and open for a full solution.