Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Research paper by A. M. Savchuk, A. A. Shkalikov

Indexed on: 30 Dec '10Published on: 30 Dec '10Published in: Functional Analysis and Its Applications

Abstract

Two inverse problems for the Sturm-Liouville operator Ly = s-y″ + q(x)y on the interval [0, fy] are studied. For θ ⩾ 0, there is a mapping F:W2θ → lBθ, F(σ) = {sk}1∞, related to the first of these problems, where W2∞ = W2∞[0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q, and lBθ is a specially constructed finite-dimensional extension of the weighted space l2θ, where we place the regularized spectral data s = {sk}1∞ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for ∥σ - σ1∥θ via the lBθ-norm ∥s − s1∥θ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case q ∈ L2, which corresponds to θ = 1.