Properties of the map associated with recovering of the Sturm-Liouville operator by its spectral function. Uniform stability in the scale of Sobolev spaces

الأصل · EN

Denote by LD the Sturm-Liouville operator Ly=-y" +q(x)y on the finite interval [0,π] with Dirichlet boundary conditions y(0)=y(π)=0. Let {λₖ}₁∞ and {αₖ}₁∞ be the sequences of the eigenvalues and norming constants of this operator. For all θ 0 we study the map F: W₂θ → lDθ defined by F(σ) ={sₖ}₁∞. Here σ= ∫ q is the primitive of q, s = {sₖ}₁∞ be regularized spectral data defined by s₂ₖ =√λₖ-k,s₂ₖ₋₁=αₖ-π/2 and lDθ are special Hilbert spaces which are constructed in the paper as finite dimensional extensions of the usual weighted l₂ spaces. We give a complete characterization of the image of this nonlinear operator, show that it is locally invertible analytic map, find explicit form of its Frechet derivative. The main result of the paper are the uniform estimates of the form σ-σ₁θ s - s₁θ, provided that the spectral data s and s₁ run through special convex sets in the spaces lDθ.