Uniform stability of the inverse Sturm–Liouville problem with respect to the spectral function in the scale of Sobolev spaces

We consider the inverse problem of recovering the potential for the Sturm–Liouville operator $Ly=-y''+q(x)y$ on the interval $[0,\pi]$ from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed $\theta\geq0$, with this problem we associate a map $F\colon W^\theta_2\to l^\theta_\mathrm D$, $F(\sigma)=\{s_k\}_1^\infty$, where $W^\theta_2= W^\theta_2[0,\pi]$ is the Sobolev space, $\sigma=\int q$ is a primitive of the potential $q\in W^{\theta-1}_2$, and $l^\theta _\mathrm D$ is a specially constructed finite-dimensional extension of the weighted space $l^\theta_2$; this extension contains the regularized spectral data $\mathbf s=\{s_k\}_1^\infty$ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference $\|\sigma-\sigma_1\|_\theta$ in terms of the $l^\theta_\mathrm D$ norm of the difference of the regularized spectral data $\|\mathbf s-\mathbf s_1\|_\theta$. The result is new even for the classical case $q\in L_2$, which corresponds to the case of $\theta=1$.