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

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: W ₂ ^θ → l _D ^θ , F(σ) = {s _k } ₁ ^∞ , where W ₂ ^θ = W ₂ ^θ 0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W ₂ ^{θ ? 1} , and l _D ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂ ^θ ; this extension contains the regularized spectral data s = {s _k } 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 ? σ ₁‖ _θ in terms of the l _D ^θ norm of the difference of the regularized spectral data ‖s ? s₁‖ _θ . The result is new even for the classical case q ∈ L ₂, which corresponds to the case θ = 1.