# Determination of matrix potential from scattering matrix

## TL;DR

This paper proves that for matrix Schrödinger operators with rapidly decreasing potentials, the scattering matrix uniquely determines the potential and boundary conditions on the half line and the full line, with some cases relying solely on reflection coefficients.

## Contribution

It establishes uniqueness results linking the scattering matrix to the potential and boundary conditions for matrix Schrödinger operators with fast-decaying potentials.

## Key findings

- Scattering matrix uniquely determines potential and boundary condition on the half line.
- Scattering matrix or transmission/reflection coefficients determine potential on the full line.
- Left reflection coefficient alone suffices if potential vanishes on negative half-line.

## Abstract

(i) For the matrix Schr\"{o}dinger operator on the half line, it is shown that if the potential exponentially decreases fast enough then only the scattering matrix uniquely determines the self-adjoint potential and the boundary condition. (ii) For the matrix Schr\"{o}dinger operator on the full line, it is shown that if the potential exponentially decreases fast enough then the scattering matrix (or equivalently, the transmission coefficient and reflection coefficient) uniquely determine the potential. If the potential vanishes on $(-\infty,0)$ then only the left reflection coefficient uniquely determine the potential.