Non-stochasticity of time-dependent quadratic Hamiltonians and the spectra of canonical transformations

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Abstract
The authors study the quantum mechanical evolution generated by Hamiltonians which are quadratic polynomials in q and p with time-dependent coefficients. This evolution is determined by a unitary implementation of the phase flow of the corresponding classical Hamiltonian. In the case of quadratic Hamiltonians which are periodic in time, the Floquet operator is shown to have either a pure point spectrum or a purely transient absolutely continuous spectrum. Thus, the motion is non-stochastic. In a simple model of a quadratic Hamiltonian with random time dependence, the quantum mechanical motion is shown to be non-stochastic almost surely.

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