A QUANTIZATION CONDITION AND ENERGY SHIFT FOR WEAK CHAOS

A mechanism of quantizing chaos is discussed and an analytical quantiz ation condition for weak chaos is derived. In the asymptotic evaluatio n of the density of states, we include the effect of the rapid change of the amplitude factor in the Feynman kernel, whereas the Gutzwiller trace formula considers only. the violent oscillation of the usual qua ntum-phase represented by the action integral. Instead of the intracta ble application of the steepest descent method, we extract essential i nformation from the periodic-orbit theory based on the smooth relation ship between the steepest descent method and the stationary phase appr oximation. For weak chaos is set a quantization condition that detects which periodic orbits are supposed to correlate with the quantizing s teepest-descent-solutions. It is shown that the true energy to be quan tized is shifted from that of such periodic orbits. The energy thus qu antized has the same form as the Helmholtz free energy within the fram ework of our thermodynamic characterization of quantum chaos. As the i nstability disappears, this quantization condition is correctly reduce d to the resonant quantization condition, through which it is connecte d with the Einstein-Brillouin-Keller conditions.