An Analytical Explanation of Quantum
Stochastic Webs

In subsection 4.1.2 unbounded diffusive motion has been shown to exist for the quantum kicked harmonic oscillator in the classical resonance cases (1.23) defined by $q\in\tilde{{\mathcal Q}}$ as given by equation (1.33). What is more, the quantum phase portraits in subsection 4.1.1 clearly demonstrate that the unbounded quantum motion takes place in just the channels of the classical periodic stochastic webs discussed in section 1.2. In other words, it has been found numerically that there exist quantum stochastic webs covering the whole phase plane and exhibiting the same symmetries as their classical counterparts described in chapter 1.

In this section 4.2 I review, along the lines of [BR95], an analytical explanation for these observations, using an argument that relies on exploiting the symmetries of the FLOQUET operator and on constructing groups of mutually commuting translation operators in the phase plane that also commute with the FLOQUET operator. A related, though slightly less transparent, line of reasoning may be found in [BRZ91]. The cases $q=1$ and $q=2$, belonging to ${\mathcal Q}$ -- see equation (1.30) -- but not to $\tilde{{\mathcal Q}}$, are discussed here, too, as well as the remaining resonances with $q\not\in {\mathcal Q}$.