Recovering of a potential of Sturm-Liouville operator from a finite sets of eigenvalues and norming constants

Abstract

It is well known that a potential $q$ of the Sturm-Liouville operator $Ly=-y"+q(x)y$ on the finite interval $[0,\pi ]$ can be uniquely recovered by the spectrum $\{{\lambda }_k{\}}_1^{\infty }$ and norming constants $\{{\alpha }_k{\}}_1^{\infty }$ of this operator with Dirichlet boundary conditions. Given potential $q$ belonging to Sobolev space $W_2^{\theta }[0,\pi ]$ with $\theta >-1$ we associate its $2N$-approximation $q_N$ constructed by the final sets $\{{\lambda }_k{\}}_1^N$ and $\{{\alpha }_k{\}}_1^N$. The main result claims that for $-1⩽\tau <\theta$ the estimate $\Vert q-q_N{\Vert }_{\tau }⩽C{N}^{\theta -\tau }$ holds, where $\Vert \cdot {\Vert }_{\tau }$ is the norm in $W_2^{\tau }$ and the constant $C$ depends on $R$ but does not depend on $q$ if $\Vert q{\Vert }_{\theta }⩽R$.

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