Non-stochasticity of time-dependent quadratic Hamiltonians and the spectra of canonical transformations. (English) Zbl 0601.70013

Hagedorn, George A.; Loss, Michael; Slawny, Joseph

J. Phys. A 19, 521-531 (1986).

The authors consider a classical Hamiltonian which is a quadratic polynomial in q and p with time-dependent coefficients. By assuming that the time dependence of the coefficients is piecewise continuous, the authors prove that the Floquet operator has either a strictly pure point spectrum or has a strictly transient absolutely continuous spectrum. This is supposed to imply that the corresponding quantum mechanical motion is non-stochastic. Further, by considering a simple model of a quadratic Hamiltonian with random time-dependence, the authors show that the corresponding quantum mechanical motion is almost surely non-stochastic.

Reviewer: Ch.Sharma

MSC:

70H25	Hamilton’s principle
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70H05	Hamilton’s equations

Keywords:

classical Hamiltonian; quadratic polynomial; Floquet operator; strictly pure point spectrum; strictly transient absolutely continuous spectrum; corresponding quantum mechanical motion; quadratic Hamiltonian with random time-dependence