On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

By a straightforward generalization, we extend the work of Combescure [J. Stat. Phys. 59, 679 (1990)] from rank-1 to rank-N perturbations. The requirement for the Floquet operator to be pure point is established and compared to that in Combescure. The result matches that in McCaw and McKeller [J. Math. Phys. 46, 032108 (2005)]. The method here is an alternative to that work. We show that if the condition for the Floquet operator to be pure point is relaxed, then in the case of the delta-kicked Harmonic oscillator, a singularly continuous component of the Floquet operator spectrum exists. We also provide an in-depth discussion of the conjecture presented in the work of Combescure of the case where the unperturbed Hamiltonian is more general. We link the physics conjecture directly to a number-theoretic conjecture of Vinogradov [The Method of Trigonometrical Sums in the Theory of Numbers (Interscience, London, 1954)] and show that a solution of Vinogradov's conjecture solves the physics conjecture. The result is extended to the rank-N case. The relationship between our work and the work of Bourget [J. Math. Anal. Appl. 276, 28 (2002); 301, 65 (2005)], on the physics conjecture is discussed. (c) 2005 American Institute of Physics.