Dynamics and quantization of Hamiltonian chaos: Density of states in phase-space semiclassical mechanics

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Abstract
We derive an extended expression of the density of states for Hamiltonian chaos in the present paper and will discuss the possibility of explicit construction of the quantization condition for chaos in a future article. In view of the controversial validity of Gutzwiller’s density of states, we apply our phase-space semiclassical mechanics [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988); Phys. Rev. A 39, 5961 (1989)] in order to construct independently the expression for both regular and irregular spectra in a unified manner. It is shown that the Liapunov exponent, Greene’s residue in classical chaos, and the Maslov index in the Einstein-Brillouin-Keller conditions are closely related to each other through the amplitude factor of our phase-space kernel. In particular, the Maslov index is interpreted as a quantum-mechanical phase due to the spinning motion of a volume element that is to be carried by a phase flow along a periodic orbit. The difficulty in semiclassical mechanics is argued in terms of the exponential decay of a time correlation function. Furthermore, a formal aspect of the thermodynamic characterization of quantum chaos is addressed, defining the dynamical temperature and entropy through the analytic structure of the density of states.

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