Scattering from the Potential Barrier V=cosh^{-2} \omega x from the Path Integration over SO(1,2)

TL;DR

This paper derives the unitary irreducible representations of SO(1,2), develops addition theorems for path integrals, and evaluates the Green function and scattering coefficients for a hyperbolic potential barrier, including moving barriers.

Contribution

It introduces new addition theorems for SO(1,2) representations and computes the Green function for the cosh^{-2} potential using path integrals over the coset space.

Findings

Green function for the potential barrier is explicitly evaluated.

Transition and reflection coefficients are derived.

Results for moving barriers are also obtained.

Abstract

Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and noncompact basis and the new addition theorems are derived which are required in path integral applications involving the positively signed potential. The Green function for the potential barrier $V=cosh^{-2}\omega x$ is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup. The transition and the reflection coefficients are given. Results for the moving barrier $V=\cosh^{-2}\omega (x-g_0t)$ are also presented.