Generalized Floquet Theory: Application to Dynamical Systems with Memory and Bloch's Theorem for Nonlocal Potentials

Generalized Floquet Theory: Application to Dynamical Systems with Memory and Bloch’s Theorem for Nonlocal Potentials

Fabio L. Traversa, Massimiliano Di Ventra, and Fabrizio Bonani

Phys. Rev. Lett. 110, 170602 – Published 24 April 2013

Abstract

Floquet theory is a powerful tool in the analysis of many physical phenomena, and extended to spatial coordinates provides the basis for Bloch’s theorem. However, in its original formulation it is limited to linear systems with periodic coefficients. Here, we extend the theory by proving a theorem for the general class of systems including linear operators commuting with the period-shift operator. The present theorem greatly expands the range of applicability of Floquet theory to a multitude of phenomena that were previously inaccessible with this type of analysis, such as dynamical systems with memory. As an important extension, we also prove Bloch’s theorem for nonlocal potentials.

Received 11 December 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.170602

Authors & Affiliations

Fabio L. Traversa^1,2,*, Massimiliano Di Ventra^2,†, and Fabrizio Bonani^3,‡

¹Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193-Bellaterra, Barcelona, Spain
²Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
³Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, 10129 Torino, Italy