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 $W^\theta_2[0, \pi]$ 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 $-1\leqslant\tau <\theta$ the estimate $\|q -q_N\|_\tau \leqslant CN^{\theta-\tau}$ holds, where $\|\cdot\|_\tau$ is the norm in $W^\tau_2$ and the constant $C$ depends on $R$ but does not depend on $q$ if $\|q\|_\theta \leqslant R$.