Analytical solution of the Feynman Kernel for general exponential-type potentials

Abstract

This paper presents an analytical path-integral treatment of the ℓ-states of an exponential-type potential. We propose a generalization of the Pekeris approximation of the centrifugal term adapted to deformed potentials. To obtain solutions of the radial Feynman Kernel for arbitrary angular number, we perform a nontrivial change of variable accompanied by a local time rescaling. Using Euler angles and the isomorphism between S3 and SU(2), we convert the radial path integral into a maniable one. Analytical expressions of the energy spectrum and the normalized ℓ-state eigenfunctions are derived from the Green function. Several potentials are obtained as special cases of the general exponential-type potential. Thus, their eigenvalues and eigenfunctions are deduced straightforwardly. Numerical results show that our technique improves the state-of-the-art.