On the Properties of Maps Connected with Inverse Sturm--Liouville Problems

A. M. Savchuk ; A. A. Shkalikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 227-247

Résumé

Let $L_\mathrm D$ be the Sturm–Liouville operator generated by the differential expression $Ly=-y''+q(x)y$ on the finite interval $[0,\pi]$ and by the Dirichlet boundary conditions. We assume that the potential $q$ belongs to the Sobolev space $W^\theta_2[0,\pi]$ with some $\theta\geq-1$. It is well known that one can uniquely recover the potential $q$ from the spectrum and the norming constants of the operator $L_\mathrm D$. In this paper, we construct special spaces of sequences $\widehat l_2^{\,\theta}$ in which the regularized spectral data $\{s_k\}_{-\infty}^\infty$ of the operator $L_\mathrm D$ are placed. We prove the following main theorem: the map $Fq=\{s_k\}$ from $W^\theta _2$ to $\widehat l_2^{\,\theta}$ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator $L_\mathrm{DN}$ generated by the same differential expression and the Dirichlet–Neumann boundary conditions. These results serve as a basis for solving the problem of uniform stability of recovering a potential. Note that this problem has not been considered in the literature. The uniform stability results are formulated here, but their proof will be presented elsewhere.