Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 8

On the positivity of trace class operators

Pages: 2061 – 2091

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a4

Authors

Elena Cordero (Dipartimento di Matematica, Università di Torino, Italy)

Maurice de Gosson (Faculty of Mathematics, Universität Wien, Austria)

Fabio Nicola (Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy)

Abstract

The characterization of positivity properties of Weyl operators is a notoriously difficult problem, and not much progress has been made since the pioneering work of Kastler, Loupias, and Miracle-Sole (KLM). In this paper we begin by reviewing and giving simpler proofs of some known results for trace-class Weyl operators; the latter play an essential role in quantum mechanics. We then apply time-frequency analysis techniques to prove a phase space version of the KLM condition; the main tools are Gabor frames and the Wigner formalism. Finally, discrete approximations of the KLM condition, which are tractable numerically, are provided.

Maurice de Gosson was funded by the Grant P27773 of the Austrian Research Foundation FWF. Elena Cordero and Fabio Nicola were partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).