  The focus of this work is a correspondence between the Hilbert space
operators on one hand, and doubly periodic generalized functions on the other.
The linear map that implements it, referred to as the Q-transform, enables a
direct application of the classical Harmonic Analysis in a study of quantum
systems. In particular, the Q-transform makes it possible to reinterpret the
dynamic of a quantum observable as a (typically nonlocal) dynamic of a
classical observable. From this point of view we carry out an analysis of an
open quantum system whose dynamics are governed by an asymptotically harmonic
Hamiltonian and compact type Lindblad operators. It is established that the
initial value problem of the equivalent nonlocal but classical evolution is
well posed in the appropriately chosen Sobolev spaces.
  The second set of results pertains to a generalization of the basic
Q-transform and highlights a certain type of asymptotic redundancy. This
phenomenon, referred to as the broadband redundancy, is a consequence of a
well-known property of the zeros of the Riemann zeta function, namely, the
uniform distribution modulo one of their ordinates. Its relevance to the
analysis of quantum dynamics is only a special instance of its utility in
harmonic analysis in general. It remains to be seen if the phenomenon is
significant also in the physical sense, but it appears well-justified---in
particular, by the results presented here---to pose such a question.
