A nonlocal transform to map and track quantum dynamics

Abstract

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 doubly periodic real function. 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.