Decay of quasibound states of multidimensional systems with a barrier: A semiclassical transfer matrix approach

We present a semiclassical method for calculating the positions and widths of tunneling resonances in multi-dimensional systems with a potential energy barrier. The treatment is applicable to arbitrary resonance states with no restrictions concerning anharmonicity, integrability, or resonance overlap. At energies below the barrier, the method is based on the choice of a particular surface that divides phase space into two regions, one including a potential energy well and the other including the barrier. Transfer matrices are constructed for each region from short, real-valued, classical trajectories confined to a single zone. Transitions between the regions are described by forming products of such matrices. These matrices are used to form the Green function for the system, and resonance positions and energies can be obtained, in principle, from its poles at complex energies. In practice, these resonance parameters are determined from simple formulas at real energies. The avoidance of general complex-valued trajectories in this approach greatly simplifies calculations. At energies above the barrier, we construct the transfer matrix for the well region from classical trajectories that travel to and from compound dividing surfaces. These combine surfaces at which these trajectories are classically reflected from the barrier with those at which they are classically transmitted across the barrier. Numerical results for model systems are presented.