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Recent Advances in the Theory of Bogoliubov Hamiltonians

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Macroscopic Limits of Quantum Systems (MaLiQS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 270))

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Abstract

Bosonic quadratic Hamiltonians, often called Bogoliubov Hamiltonians, play an important role in the theory of many-boson systems where they arise in a natural way as an approximation to the full many-body problem. In this note, we would like to give an overview of recent advances in the study of bosonic quadratic Hamiltonians. In particular, we relate the reported results to what can be called the time-dependent diagonalization problem.

Dedicated to Herbert Spohn on the occasion of his 70th birthday

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Notes

  1. 1.

    If \(C:\mathfrak {h}\rightarrow \mathfrak {K}\) is anti-linear, then \(C^*: \mathfrak {K} \rightarrow \mathfrak {h}\) is defined by \(\langle C^*g,f \rangle _{\mathfrak {h}} = \langle C f, g \rangle _{\mathfrak {K}}\) for all \(f\in \mathfrak {h},g\in \mathfrak {K}.\) The anti-linear map C is an anti-unitary if \(C^*C=1_{\mathfrak {h}}\) and \(CC^*=1_{\mathfrak {K}}\).

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Acknowledgements

The support of the National Science Centre (NCN) project Nr. 2016/21/D/ST1/02430 is gratefully acknowledged.

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Authors and Affiliations

  1. Faculty of Physics, Department of Mathematical Methods in Physics, University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland

    Marcin Napiórkowski

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  1. Marcin Napiórkowski
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Correspondence to Marcin Napiórkowski .

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Editors and Affiliations

  1. Institut für Theoretische, Physik Universität Leipzig, Leipzig, Germany

    Daniela Cadamuro

  2. Zentrum Mathematik, M5, Technische Universität München, Garching, Germany

    Maximilian Duell

  3. Zentrum Mathematik, M5, Technische Universität München, Garching, Germany

    Wojciech Dybalski

  4. UMPA UMR 5669 CNRS, École normale supérieure, Lyon, France

    Sergio Simonella

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Napiórkowski, M. (2018). Recent Advances in the Theory of Bogoliubov Hamiltonians. In: Cadamuro, D., Duell, M., Dybalski, W., Simonella, S. (eds) Macroscopic Limits of Quantum Systems. MaLiQS 2017. Springer Proceedings in Mathematics & Statistics, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-030-01602-9_5

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