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

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.