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

TL;DR

This paper extends the analysis of the spectral properties of Floquet operators in periodically kicked quantum systems, exploring conditions for pure point spectra and the existence of singular continuous components, with connections to number theory.

Contribution

It generalizes previous work from rank-1 to rank-N perturbations and links spectral properties to a number-theoretic conjecture, offering an alternative method for analysis.

Findings

Floquet operator is pure point under certain conditions

Existence of singular continuous spectrum in delta-kicked harmonic oscillator

Connection between spectral properties and Vinogradov's conjecture

Abstract

By a straightforward generalisation, we extend the work of Combescure 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. 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 Combescure of the case where the unperturbed Hamiltonian is more general. We link the physics conjecture directly to a number-theoretic conjecture of Vinogradov 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 on the physics conjecture is discussed.