| The Green's function and its trace are expressed in terms of a semiclassical initial value approach. The resulting formulas do not require determination of special root trajectories or periodic orbits and are valid irrespective of the chaotic or integrable nature of the classical dynamics. The quantities of interest are obtained by integrating over points on Poincare surfaces which serve as initial values for classical trajectories. The values of the classical variables at subsequent intersections are used to form the integrands. This treatment is especially advantageous for systems with homogeneous potential energy functions since it allows one to exploit the energy scaling of the classical dynamics in a very efficient way. A test calculation of the energy eigenvalues for a classically chaotic two-dimensional quartic oscillator are described. For this case it is found that that the present method converges about 50 times more quickly than one based on a more conventional time-dependent initial value treatment. Other applications of this Green's function treatment, including initial value formulations of the transfer matrix and the semiclassical zeta function, are also discussed. |
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