Abstract
Density operators are positive semidefinite operators with trace one representing the mixed states of quantum mechanics. The purpose of this contribution is to define and study a subclass of density operators on \(L^{2}(\mathbb {R}^{n})\), which we call Toeplitz density operators. They correspond to quantum states obtained from a fixed function (“window”) by position-momentum translations, and reduce in the simplest case to the anti-Wick operators considered long ago by Berezin and extensively studied by Cordero and others. The rigorous study of Toeplitz operators requires the use of classes of functional spaces defined by Feichtinger.
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Acknowledgements
This work was written while the author was holding the Giovanni Prodi visiting chair at the Julius-Maximilians-Universität Würzburg during the summer semester 2019. It is my pleasure to thank the referee for useful comments and suggestions.
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Gosson, M.d. (2020). Generalized Anti-Wick Quantum States. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-56005-8_7
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