Inverse Problems for Sturm–Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability

Two inverse problems for the Sturm–Liouville operator $Ly=-y''+q(x)y$ on the interval $[0,\pi]$ are studied. For $\theta\ge0$, there is a mapping $F\colon W^{\theta}_2 \to l^{\theta}_B$, $F(\sigma)=\{s_k\}_1^\infty$, related to the first of these problems, where $W^\theta_2= W^{\theta}_2[0,\pi]$ is the Sobolev space, $\sigma =\int q$ is a primitive of the potential $q$, and $l^{\theta}_B$ is a specially constructed finite-dimensional extension of the weighted space $l^{\theta}_2$, where we place the regularized spectral data ${\mathbf s}=\{s_k\}_1^\infty$ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for $\|\sigma - \sigma_1\|_\theta$ via the $l^{\theta}_B$-norm $\|{\mathbf s}-{\mathbf s}_1\|_\theta$ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator $L$ generated by the Dirichlet boundary conditions. The result is new even for the classical case $q\in L_2$, which corresponds to $\theta =1$.