Semiclassical quantization of KAM resonances in time-periodic systems

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Abstract
A semiclassical theory for the quasi-energy spectrum of time-periodic systems with accidental classical resonances is presented. The primitive EBK quantum conditions for integrable systems are extended to multiply periodic flux tubes occurring in resonant systems. Replacing classical actions by appropriate differential operators in a classical resonance Hamiltonian yields a uniform quantization of states related to a classical resonance region. The derivation being general for time-periodic systems unfolds the organization of the quasi-energy spectrum, reducing it to the spectrum of a single time-independent Hamiltonian of one degree of freedom with additional rational shifts of h(cross) omega . In a first-order approximation the resonance Hamiltonian is reduced to a pendulum leading to a differential equation of the Mathieu type for the quasi-energies. It is rigorously shown how parameters of the differential equation can be drawn from classical dynamics, using the data of the 'essential' orbits in the resonance zone, i.e. stability coefficients and actions of hyperbolic and elliptic orbits as well as actions of homoclinic orbits. The quasi-energy spectrum of a forced quartic oscillator is studied numerically and evaluated. Semiclassical quasi-energies related to a resonance of period three are computed and compared with exact quantum mechanical eigenvalues.