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 $[0,\pi]$. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map $F:\, W^\theta_2\to l_B^\theta,\ F(\sigma) =\{s_k\}_1^\infty$, where $W^\theta_2 = W^\theta_2[0,\pi]$ are Sobolev spaces with $\theta\geqslant 0$, $\sigma=\int q$ is a primitive of the potential $q$ and $l_B^\theta$ 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 $\|\sigma -\sigma_1\|_\theta$ by the norm of the difference of the regularized spectral data $\|\bold s -\bold s_1\|_\theta$ where the last norm is taken in $l_B^\theta$. 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 $\theta =1$.