On the properties of maps connected with inverse Sturm-Liouville problems

Let L _D be the Sturm-Liouville operator generated by the differential expression L _y = ?y″ + q(x)y on the finite interval 0, π] and by the Dirichlet boundary conditions. We assume that the potential q belongs to the Sobolev space W ₂ ^? 0, π] with some ? ≥ ?1. It is well known that one can uniquely recover the potential q from the spectrum and the norming constants of the operator L _D. In this paper, we construct special spaces of sequences ? ₂ ^θ in which the regularized spectral data {s _k } _?∞ ^∞ of the operator L _D are placed. We prove the following main theorem: the map F _q = {s _k } from W ₂ ^? to ? ₂ ^θ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator L _DN generated by the same differential expression and the Dirichlet-Neumann boundary conditions. These results serve as a basis for solving the problem of uniform stability of recovering a potential. Note that this problem has not been considered in the literature. The uniform stability results are formulated here, but their proof will be presented elsewhere.