Recovering the M-channel Sturm-Liouville operator from M+1 spectra

TL;DR

This paper establishes a method to uniquely recover the potential matrix in a multichannel Schrödinger equation using spectral data and norming constants, advancing inverse spectral theory for coupled quantum systems.

Contribution

It introduces a novel relationship between norming constants and spectra for M coupled Schrödinger equations, enabling unique potential reconstruction from spectral data.

Findings

Unique recovery of potential matrix from spectral data.

Explicit relationship between norming constants and spectra.

Method applicable to systems with specific boundary conditions.

Abstract

For a system of M coupled Schroedinger equations, the relationship is found between the vector-valued norming constants and M+1 spectra corresponding to the same potential matrix but different boundary conditions. Under a special choice of particular boundary conditions, this equation for norming vectors has a unique solution. The double set of norming vectors and associated spectrum of one of the M+1 boundary value problems uniquely specifies the matrix of potentials in the multichannel Schroedinger equation.