2010/10/28 by A. M. Savchuk, Savchuk, A. M., A. A. Shkalikov +1 · 2 citations
Mathematics · #34L40 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:34L40
paper · pdf · doi:10.48550/arxiv.1010.5916
21 pages
arxiv created 2010/10/28 · arxiv updated 2010/10/29
The paper deals with two inverse problems for Sturm--Liouville operator Ly=-y" +q(x)y on the finite interval [0,π]. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map F: Wθ2→ lBθ, F(σ) =\sk\1^∞, where Wθ2 = Wθ2[0,π] are Sobolev spaces with θ\geqslant 0, σ=∫ q is a primitive of the potential q and lBθ are special Hilbert spaces which we construct to place in the regularized spectral data \bold s = \sk\1^∞. The properties of the map F are studied in details. The main result is the theorem on uniform stability. It gives uniform estimates from above and below of the norm of the difference ‖σ-σ1‖θ by the norm of the difference of the regularized spectral data ‖\bold s -\bold s1‖θ where the last norm is taken in lBθ. A similar result is obtained for the second inverse problem when the potential is recovered by the spectral function of the operator L generated by Dirichlet boundary conditions. The results are new for classical case q∈ L2 which corresponds to the value θ=1.