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 RSs used in the previous RS expansion Born approximation or resonant-state expansion (RSE) 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 85 023835; Doost et al 2013 Phys. Rev. A 87 043827; Armitage et al 2014 Phys. Rev. A 89; Doost et al 2014 Phys. Rev. A 90 013834) and normalised correctly to appear in 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), Doost (2016 Phys. Rev. A 93 023835). In the case of effectively one-dimensional systems I find a RSE Born approximation alternative to the scattering matrix method.
Highlights
Fundamental to scattering theory, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point inside the scattering potential, it was first discovered by Max Born and presented in Ref. [1]
We find that the resonant-state expansion (RSE) Born approximation requires less basis states to reach a required accuracy than the spectral Green’s functions (GFs) and unlike the spectral GF is convergent outside the system
I have demonstrated that once the correct normalisation is used in the RSE Born approximation convergences towards the exact solution is obtained
Summary
Fundamental to scattering theory, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point inside the scattering potential, it was first discovered by Max Born and presented in Ref. [1]. The numerical study made in Ref.[2] only included a single RS in the expansion of the Born approximation, most likely to avoid divergence caused by their incorrect normalisation of the RSs. Recently there has been developed [3, 8, 9] a rigorous perturbation theory called resonant-state expansion (RSE) which was applied to one-dimensional (1D), 2D and 3D systems [4,5,6,7, 11, 12] which only calculates the modes and makes no use of them. Determining the effect of perturbations which break the symmetry presents a significant challenge as these popular computational techniques need large computational resources to model high quality modes These methods generate spurious solutions which would damage the accuracy of the RSE Born approximation if included in the basis. V gives the numerical validation of the new method along with a comparison of the alternative RSE approaches
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