Ali Mostafazadeh
On some recent developments in scattering theory: Fundamental transfer matrix and exactness of N-th order Born approximation
Potential scattering admits a formulation in terms of a fundamental notion of transfer matrix. This is a linear operator possessing a Dyson series expansion for an effective non-Hermitian Hamiltonian operator. This approach to potential scattering has so far led to several interesting developments. The most notable are the construction of the first examples of short-range potentials for which the N-th order Born approximation is exact, potentials that display broadband directional invisibility, and a singularity-free treatment of delta-function potentials lying on a line in two dimensions and on a plane in three dimensions. It has also been generalized to electromagnetic scattering and used to deal with certain electromagnetic radiation problems. This talk presents a brief review of these developments and addresses the mathematical problem of the existence of the fundamental transfer matrix within the context of propagating-wave approximation in two dimensions. 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. The latter includes an infinite subclass of potentials for which the N-th order Born approximation is exact with N depending on the frequency of the incident wave.
