Inverse problems for Sturm--Liouville operators with potentials from Sobolev spaces. Uniform stability

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 $F:\, W^θ_2\to l_B^θ,\ F(σ) =\{s_k\}_1^\infty$, where $W^θ_2 = W^θ_2[0,π]$ are Sobolev spaces with $θ\geqslant 0$, $σ=\int q$ is a primitive of the potential $q$ and $l_B^θ$ are special Hilbert spaces which we construct to place in the regularized spectral data $\bold s = \{s_k\}_1^\infty$. 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 $\|σ-σ_1\|_θ$ by the norm of the difference of the regularized spectral data $\|\bold s -\bold s_1\|_θ$ where the last norm is taken in $l_B^θ$. 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 $q\in L_2$ which corresponds to the value $θ=1$.

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