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Square-integrability of multivariate metaplectic wave-packet representations. (English) Zbl 1362.81046

Summary: This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space \({{L}^{2}}\left( \mathbb{R}^{d} \right)\). The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group \(\mathrm{Sp}\left({{\mathbb{R}}^{d}}\right)\), which guarantees the square-integrability of the associated metaplectic wave-packet representation on \({{L}^{2}}\left({{\mathbb{R}}^{d}}\right)\).

MSC:

81R30 Coherent states
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
53D22 Canonical transformations in symplectic and contact geometry
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