# Potentials with Identical Scattering Properties Below a Critical Energy

## TL;DR

This paper introduces a criterion for determining when two complex scattering potentials in two or three dimensions are equivalent in their scattering properties below a certain energy, using a multidimensional transfer-matrix approach.

## Contribution

It provides a simple criterion for alpha-equivalence of complex potentials in higher dimensions based on a new transfer-matrix formulation.

## Key findings

- Derived a criterion for alpha-equivalence in 2D and 3D scattering potentials.
- Applied the transfer-matrix approach to identify potentials with identical low-energy scattering.
- Enhanced understanding of potential equivalence in multidimensional scattering theory.

## Abstract

A pair of scattering potentials are called $\alpha$-equivalent if they have identical scattering properties for incident plane waves with wavenumber $k\leq\alpha$ (energy $k^2\leq\alpha^2$.) We use a recently developed multidimensional transfer-matrix formulation of scattering theory to obtain a simple criterion for $\alpha$-equivalence of complex potentials in two and three dimensions.