Inverse spectral problems for Sturm--Liouville operators with matrix-valued potentials

TL;DR

This paper characterizes the spectral data for matrix-valued Sturm--Liouville operators with potentials in a Sobolev space and proposes an algorithm to reconstruct the potential from this data using Krein's method.

Contribution

It provides a complete description of spectral data and introduces a reconstruction algorithm for matrix-valued potentials in the Sobolev space $W_2^{-1}$.

Findings

Complete spectral data characterization for matrix-valued potentials.

An explicit reconstruction algorithm based on Krein's accelerant method.

Applicability to Sturm--Liouville operators on [0,1].

Abstract

We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.