  The general treatment of a separable Hamiltonian of Liouville-type is
well-known in operator formalism. A path integral counterpart is formulated if
one starts with the Jacobi's principle of least action, and a path integral
evaluation of the Green's function for the hydrogen atom by Duru and Kleinert
is recognized as a special case. The Jacobi's principle of least action for
given energy is reparametrization invariant, and the separation of variables in
operator formalism corresponds to a choice of gauge in path integral. The
Green's function is shown to be gauge independent,if the operator ordering is
properly taken into account. These properties are illustrated by evaluating an
exact path integral of the Green's function for the hydrogen atom in parabolic
coordinates.
