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 ϑ [0, π] with some ϑ ≥ −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 ɫ 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 ϑ to ɫ 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.

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