Resonant-state expansion Born Approximation with a correct eigen-mode normalisation applied to Schrodinger's equation or general wave equations

TL;DR

This paper introduces the RSE Born Approximation for Schrödinger's equation, utilizing resonant states with correct normalization, which converges to the exact solution and offers an alternative to traditional scattering matrix methods in 1D systems.

Contribution

It applies the RSE Born Approximation to Schrödinger's equation with proper eigen-mode normalization, demonstrating convergence and providing a new approach for 1D scattering problems.

Findings

Converges to the exact solution with infinite resonant states

Allows calculation of strong scattering at all frequencies

Provides an alternative to scattering matrix in 1D systems

Abstract

The RSE Born Approximation is a new scattering formula in Physics, it allows the calculation of strong scattering at all frequencies via the Fourier transform of the scattering potential and Resonant-states. In this paper I apply the RSE Born Approximation to Schrodinger's equation. The resonant-states of the system can be calculated using the recently discovered RSE perturbation theory and correctly normalised to appear in spectral Green's functions via the flux-volume normalisation. In the limit of an infinite number of resonant states being used in the RSE Born Approximation basis the approximation converges to the exact solution. In the case of effectively 1-dimensional systems I find an RSE Born Approximation alternative to the scattering matrix method.