Tunneling And The Semiclassical Spectrum For An Isolated Classical Resonance

The island tori, surrounding a stable periodic orbit in a classical resonance, are not quantized by the Bohr-Sommerfeld rules, since these take no account of tunneling. The approximate four-dimensional Hamiltonian, generating the motion near the resonance, is integrable and can be separated by a canonical transformation. The resulting single degree of freedom problem has no momentum (action) symmetry, but it can still be mapped onto the ordinary Schrödinger equation with Bloch (Floquet) boundary conditions. The canonical invariance of the real and imaginary classical actions, which determine the semiclassical tunneling transmission coefficient, results in an eigenenergy condition which is uniformly valid for both the island tori and the tori which envelop the resonance. Tunneling between the latter is seen to be the cause of "avoided crossings" of pairs of energy levels. Different degrees of approximation are computationally verified for a model Hamiltonian whose eigenvalues can be calculated exactly. © 1984 American Chemical Society.88256139614