Multi-channel scattering problems: Analytical approach for exact solution using Green's functions

TL;DR

This paper presents a general analytical method for solving multi-channel scattering problems with Dirac Delta couplings, enabling simplified calculations of transition probabilities using Green's functions.

Contribution

It introduces a novel analytical approach that reduces multi-channel problems to independent single-channel problems using Green's functions, applicable to any potentials with known uncoupled Green's functions.

Findings

Derived a simple formula for transition probabilities between channels.

Validated the approach for arbitrary potentials with known Green's functions.

Provided a framework for exact solutions in complex scattering scenarios.

Abstract

We have proposed an analytical approach for exact solution of multi-channel scattering problems, in presence of Dirac Delta function couplings. Our solution is quite general and is valid for any set of potentials, if the Green's functions of the uncoupled potentials are known at the crossing point. Using our model, it is possible to express N-channel problem by N independent `single' channel problems and hence one can have a realistic solution of multi-channel scattering problem. In this paper we have shown that transition probability from first diabatic potential to any other potential can be easily evaluated using a very simple analytical formula - which only require value of eigenfunction of first uncoupled potential and value of Green's function of other uncoupled potential, at the crossing.