Properties of the map associated with recovering of the Sturm-Liouville operator by its spectral function. Uniform stability in the scale of Sobolev spaces

TL;DR

This paper characterizes the map between the potential of a Sturm-Liouville operator and its spectral data in Sobolev spaces, establishing uniform stability and invertibility properties.

Contribution

It provides a complete description of the spectral data map's image, its local invertibility, and explicit derivatives, with uniform stability estimates in Sobolev spaces.

Findings

The spectral data map is a locally invertible analytic operator.

Explicit form of the Frechet derivative of the spectral data map.

Uniform stability estimates in Sobolev spaces for spectral data and potential.

Abstract

Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$. Let $\{\lambda_k\}_1^\infty$ and $\{\alpha_k\}_1^\infty$ be the sequences of the eigenvalues and norming constants of this operator. For all $\theta \geqslant 0$ we study the map $F: W_2^{\theta} \to l_D^\theta$ defined by $F(\sigma) =\{s_k\}_1^\infty$. Here $\sigma= \int q $ is the primitive of $q$, $\bold s = \{s_k\}_1^\infty$ be regularized spectral data defined by $s_{2k} =\sqrt{\lambda_k}-k,\ s_{2k-1}=\alpha_k-\pi/2$ and $l_D^\theta$ are special Hilbert spaces which are constructed in the paper as finite dimensional extensions of the usual weighted $l_2$ spaces. We give a complete characterization of the image of this nonlinear operator, show that it is locally invertible analytic map, find explicit form of its Frechet derivative. The main result of the paper are the uniform estimates of the form $\|\sigma-\sigma_1\|_\theta \asymp \|\bold s -\bold s_1\|_\theta$, provided that the spectral data $\bold s$ and $\bold s_1$ run through special convex sets in the spaces $l_D^\theta$.