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:W θ2 → l θ B , F(σ) = {s k } ∞1 , related to the first of these problems, where W ∞2 = W ∞2 [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q, and l θ B is a specially constructed finite-dimensional extension of the weighted space l θ2 , where we place the regularized spectral data s = {s k } ∞1 in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for ∥σ - σ1∥θ via the l θ B -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 ∈ L 2, which corresponds to θ = 1.
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Translated from Funktsionals’nyi Analiz i Ego Prilozheniya, Vol. 44, No. 4, pp. 34-53, 2010
Original Russian Text Copyright © by A. M. Savchuk and A. A. Shkalikov
Supported by RFBR grant 10-01-00423.
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Savchuk, A.M., Shkalikov, A.A. Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability. Funct Anal Its Appl 44, 270–285 (2010). https://doi.org/10.1007/s10688-010-0038-6
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DOI: https://doi.org/10.1007/s10688-010-0038-6