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

Abstract

Let LD be the Sturm-Liouville operator generated by the differential expression Ly = −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 W2ϑ[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 LD. In this paper, we construct special spaces of sequences ɫ2θ in which the regularized spectral data {sk}−∞∞ of the operator LD are placed. We prove the following main theorem: the map Fq = {sk} from W2ϑ to ɫ2θ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator LDN 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.