A semiclassical initial value approximation for the trace of Green's function

A semiclassical initial value approximation for the trace of Green's function is derived. In contrast to the well-known formula of Gutzwiller, applicability of the present expression does not require knowledge of the system's periodic orbits but constructs the trace from classical trajectories originating from all points on a Poincaré surface. A given trajectory provides a contribution to the trace each time it returns to the surface with a weight based, in part, on the inner product (on this surface) of coherent states associated with the initial and returning points. The treatment is generalized to obtain a version of the initial value formula that is useful for systems having discrete symmetries. The initial value trace expression is shown to be semiclassically valid for chaotic systems by a stationary phase treatment that demonstrates its reduction to Gutzwiller's formula in the classical limit. Numerical calculations of energy eigenvalues verify the applicability of the approximation not only to chaotic systems but to integrable systems and systems with mixed phase space. The approximation presented here has numerical advantages over methods for determining the trace based on initial value treatments of the time-dependent propagator, especially for systems with homogeneous potential energy functions.