Resonant-state-expansion Born approximation with a correct eigenmode normalisation

TL;DR

This paper improves the Born approximation for weak scattering calculations by incorporating correctly normalised resonant states, leading to enhanced accuracy and stability in spectral Green's function computations.

Contribution

It introduces a properly normalised resonant-state expansion (RSE) Born approximation, correcting previous normalization issues and demonstrating its effectiveness in one-dimensional systems.

Findings

Correct normalisation of RSs enhances the accuracy of the RSE Born approximation.

The method provides an alternative to scattering matrix approaches in 1D systems.

Numerical stability is improved compared to previous implementations.

Abstract

The Born approximation (Born 1926 Z.Phys.38.802) is a fundamental result in physics, it allows the calculation of weak scattering via the Fourier transform of the scattering potential. As was done by previous authors (Ge et al 2014 New J. Phys. 16 113048) the Born approximation is extended by including in the formula the resonant-states (RSs) of the scatterer. However in this study unlike previous studies the included eigen-modes are correctly normalised with dramatic positive consequences for the accuracy of the method. The normalisation of the RSs used in the previous RSE Born approximation or resonant-state-expansion Born approximation made in Ge et al (2014 New J. Phys. 16 113048) has been shown to be numerically unstable in Muljarov et al (2014 arXiv:1409.6877) and by analytics here. The RSs of the system can be calculated using my recently discovered RSE perturbation theory for dispersive electrodynamic scatterers (Muljarov et al 2010 Europhys. Lett. 92 50010; Doost et al 2012 Phys, Rev. A 89; Doost et al 2014 Phys. Rev. A 90 013834) and normalised correctly to appear in the spectral Green's functions and hence the RSE Born approximation via the flux-volume normalisation which I recently rigorously derived in Armitage et al (2014 Phys. Rev. A 89), Doost et al (2014 Phys. Rev. A 90 013834)(2016 Phys. Rev. A 93 023835). In the case of effectively one-dimensional systems I find an RSE Born approximation alternative to the scattering matrix method.