Semi-classical analysis on H-type groups
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In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodi fferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account thefinite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
KeywordsH-type groups semi-classical pseudodifferential operators semi-classical measures Wigner transform asymptotic analysis microlocal analysis
MSC(2010)35S05 22E30 46L89
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This paper was written while Clotilde Fermanian Kammerer was visiting Technische Universität München and she thanks the members of the mathematics department of this institution for their kind hospitality, especially Caroline Lasser and Simone Warzel.
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