On the path integration for the potential barrier $V_{0}\cosh ^{-2}(\omega x)$

The propagator associated to the potential barrier $V=V_{0}\cosh ^{-2}(\omega x)$ is obtained by solving path integrals. The method of delta functionals based on canonical and other transformations is used to reduce the path integral for this potential into a path integral for the Morse potential problem. The dimensional extension technique is seen to be essential for performing the multiple integral representation of the propagator. The correctly normalized scattering wave functions and the scattering function are derived. To test the method employed, the free particle and the $\delta -$function barrier are considered as limiting cases.