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Uniform stability of the inverse Sturm-Liouville problem with respect to the spectral function in the scale of Sobolev spaces

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

Indexed on: 30 Jan '14Published on: 30 Jan '14Published in: Proceedings of the Steklov Institute of Mathematics



Abstract

We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = −y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W2θ → lDθ, F(σ) = {sk}1∞, where W2θ = W2θ[0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W2θ − 1, and lDθ is a specially constructed finite-dimensional extension of the weighted space l2θ; this extension contains the regularized spectral data s = {sk}1∞ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ − σ1‖θ in terms of the lDθ norm of the difference of the regularized spectral data ‖s − s1‖θ. The result is new even for the classical case q ∈ L2, which corresponds to the case θ = 1.