A Quantization Condition and Energy Shift for Weak Chaos

Abstract

A mechanism of quantizing chaos is discussed and an analytical quantization condition for weak chaos is derived. In the asymptotic evaluation 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 quantum-phase represented by the action integral. Instead of the intractable application of the steepest descent method, we extrC\ct essential information from the periodic-orbit theory based on the smooth relationship between the steepest descent method and the stationary phase approximation. For weak chaos is set a quantization condition that detects which periodic orbits are supposed to correlate with the quantizing steepest-descent-solutions. It is shown that the true energy to be quantized is shifted from that of such periodic orbits. The energy thus quantized has the same form as the Helmholtz free energy within the framework of our thermodynamic characterization of quantum chaos. As the instability disappears, this quantization condition is correctly reduced to the resonant quantization condition, through which it is connected with the Einstein-Brillouin-Keller conditions.