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Physics Letters A

Exact green function of a damped oscillator

V. V. Dodonov, E.V. Kurmyshev, Vladimir I. Man’ko

Published Jun 1, 1979• 8 citations
Publisher
DOI
10.1016/0375-9601(79)90510-3

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The application for the integrals of the motion of a quantum system in deriving Green function or propagator is presented. The Green function is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phase space. The exact expressions for the Green functions of the dual damped oscillators and the coupled harmonic oscillators are evaluated in co-ordinate representations. The relation between the integrals of the motion method and other methods such as Feynman path integral and Schwinger method are also presented.
A general method is developed to characterize the family of classical periodic orbits from the quantum Green's function for the two-dimensional (2D) integrable systems. A decomposing formula related to the beta function is derived to link the quantum Green's function with the individual classical periodic orbits. The practicality of the developed formula is demonstrated by numerically analyzing the 2D commensurate harmonic oscillators and integrable quantum billiards. Numerical analyses reveal that the emergence of the classical features in quantum Green's functions principally comes from the superposition of the degenerate states for 2D harmonic oscillators. On the other hand, the damping factor in quantum Green's functions plays a critical role to display the classical features in mesoscopic regime for integrable quantum billiards, where the physical function of the damping factor is to lead to the coherent superposition of the nearly degenerate eigenstates.