On the existence of the fundamental transfer matrix in potential scattering, propagating-wave approximation, and exactness of the Born approximation

Potential scattering admits a formulation in terms of a fundamental notion of transfer matrix which is a linear operator admitting a Dyson series expansion given by an effective non-self-adjoint Hamiltonian operator. This approach to potential scattering has so far led to a few interesting developments in two and three dimensions. The most notable of these are the construction of the first examples of short-range potentials for which the first order Born approximation is exact, potentials that display broadband omnidirectional or unidirectional invisibility, and a singularity-free treatment of delta-function potentials lying on a line in two dimensions and on a plane in three dimensions. This talk presents a first step towards a rigorous proof of the existence of the fundamental transfer matrix in two dimensions. It offers a solution of this problem within the context of propagating-wave approximation. This is an approximation scheme that ignores the contribution of the evanescent waves to the scattering amplitude and is valid for high energies and weak potentials. It becomes exact for a class of complex potentials which includes an infinite subclass for which the N-th order Born approximation is exact with N depending on the frequency of the incident wave.
References:

• F. Loran and A. Mostafazadeh, Phys. Rev. A 106, 032207 (2022); arXiv: 2204.05153.
• F. Loran and A. Mostafazadeh, J. Phys. A 55, 435202 (2022); arXiv: 2207.10054.
• F. Loran and A. Mostafazadeh, J. Phys. A 57, 335205 (2024); arXiv:2407.19983.