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

2010/10/26 by A. M. Savchuk, Savchuk, A. M., A. A. Shkalikov +1
Mathematics · #34L40 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:34L40

paper · pdf · doi:10.48550/arxiv.1010.5344

16 pages

arxiv created 2010/10/26 · arxiv updated 2010/10/27

Abstract

Denote by L_D 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 \λ_k\₁^∞ and \α_k\₁^∞ be the sequences of the eigenvalues and norming constants of this operator. For all θ\geqslant 0 we study the map F: W₂^θ → l_D^θ defined by F(σ) =\s_k\₁^∞. Here σ= ∫ q is the primitive of q, \bold s = \s_k\₁^∞ be regularized spectral data defined by s_2k =√(λ_k)-k, s_2k-1=α_k-π/2 and l_D^θ 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-σ₁‖_θ\asymp ‖\bold s -\bold s₁‖_θ, provided that the spectral data \bold s and \bold s₁ run through special convex sets in the spaces l_D^θ.

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