Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constants

Abstract

It is well known that the potential q of the Sturm–Liouville operator L y = −yʺ + q(x)y on the finite interval [0, π] can be uniquely reconstructed from the spectrum $$\left\{ {{\lambda _k}} \right\}_1^\infty $$ and the normalizing numbers $$\left\{ {{\alpha _k}} \right\}_1^\infty $$ of the operator L D with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space $$W_2^\theta \left[ {0,\pi } \right],\theta > - 1$$ , we construct a function q N providing a 2N-approximation to the potential on the basis of the finite spectral data set $$\left\{ {{\lambda _k}} \right\}_1^N \cup \left\{ {{\alpha _k}} \right\}_1^N$$ . The main result is that, for arbitrary τ in the interval −1 ≤ τ < θ, the estimate $${\left\| {q - \left. {{q_N}} \right\|} \right._\tau } \leqslant C{N^{\tau - \theta }}$$ is true, where $${\left\| {\left. \cdot \right\|} \right._\tau }$$ is the norm on the Sobolev space $$W_2^\tau $$ . The constant C depends solely on $${\left\| {\left. q \right\|} \right._\theta }$$ .