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Maximal covariance group of Wigner transforms and pseudo-differential operators

Authors: Nuno Costa Dias, Maurice A. de Gosson and João Nuno Prata
Journal: Proc. Amer. Math. Soc. 142 (2014), 3183-3192
MSC (2010): Primary 35S99, 35P05, 53D05; Secondary 35S05
Published electronically: June 3, 2014
MathSciNet review: 3223374
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Abstract: We show that the linear symplectic and antisymplectic transformations form the maximal covariance group for both the Wigner transform and Weyl operators. The proof is based on a new result from symplectic geometry which characterizes symplectic and antisymplectic matrices and which allows us, in addition, to refine a classical result on the preservation of symplectic capacities of ellipsoids.

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Additional Information

Nuno Costa Dias
Affiliation: Departamento de Matemática, Universidade Lusófona, Av. Campo Grande, 376, 1749-024 Lisboa, Portugal

Maurice A. de Gosson
Affiliation: Faculty of Mathematics, NuHAG, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria

João Nuno Prata
Affiliation: Departamento de Matemática, Universidade Lusófona, Av. Campo Grande, 376, 1749-024 Lisboa, Portugal

Keywords: Wigner transform, symplectic covariance, Weyl operator
Received by editor(s): October 7, 2012
Published electronically: June 3, 2014
Additional Notes: The first author was supported by a research grant from the Austrian Research Agency FWF (Projektnummer P23902-N13)
The second and third authors were supported by the research grant PTDC/MAT/099880/2008 of the Portuguese Science Foundation
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society

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