Verifying the surjective relation between symmetric potential function and its Scattering Matrix in 1D

TL;DR

This paper investigates the relationship between symmetric potential functions and their scattering matrices in 1D, demonstrating surjectivity in specific cases but disproving injectivity for arbitrary potentials.

Contribution

It establishes conditions under which the surjective relation holds and clarifies limitations regarding injectivity in 1D scattering theory.

Findings

Surjective relation holds for delta and finite square wall potentials.

Injective relation does not hold for arbitrary potentials.

Provides theoretical insights into wave function symmetry and boundary conditions.

Abstract

This research focuses on the possibility of the surjective relation between symmetric potential function and its scattering matrix in 1-dimension. The theory bases on the property of wave function symmetry and boundary conditions. This research shows the surjective relation in some particular cases, delta function potential, and finite square wall potential, and disproves the injective relation of the arbitrary potential function and its S-matrix.