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 $[0, π]$ can be uniquely recovered by the spectrum $\{λ_k\}_1^\infty$ and norming constants $\{α_k\}_1^\infty$ of this operator with Dirichlet boundary conditions. Given potential $q$ belonging to Sobolev space $W^θ_2[0, π]$ with $θ> -1$ we associate its $2N$-approximation $q_N$ constructed by the final sets $\{λ_k\}_1^N$ and $\{α_k\}_1^N$. The main result claims that for $-1\leqslantτ<θ$ the estimate $\|q -q_N\|_τ\leqslant CN^{θ-τ}$ holds, where $\|\cdot\|_τ$ is the norm in $W^τ_2$ and the constant $C$ depends on $R$ but does not depend on $q$ if $\|q\|_θ\leqslant R$.