Semiclassical treatment of bound state systems. III. A uniform stationary phase integration for the multidimensional problem

The semiclassical bound state formalism is extended to the multidimensional problem by deriving an expression for the reduced density matrix which remains finite when the charge density is calculated. We derive a uniform stationary phase integration technique by mapping the phase of the semiclassical propagator onto a function which has the same behavior as the phase of the free particle propagator, enabling us to incorporate the singularity at t = 0 into the integration. The resulting rdm has the oscillatory behavior of a Bessel function of order d/2, where d is the dimensionality, rather than that of a simple trigonometric function.