This article is cited in
28 papers
On the Properties of Maps Connected with Inverse Sturm–Liouville Problems
A. M. Savchuk,
A. A. Shkalikov M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
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.
UDC:
517.984 Received in August 2007
Fulltext:
PDF file (280 kB)