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

A. M. Savchuk ; A. A. Shkalikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 188-203

Résumé

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 $\theta=1$.