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Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian

  • Alessandro OlgiatiEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 18)

Abstract

The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the “projection counting” scheme.

Keywords

Bose-Einstein condensate Effective evolution equations Gross-Pitaevskii scaling Magnetic Gross-Pitaevskii equation Magnetic Laplacian Magnetic Sobolev space Magnetic vector potential Many-body quantum dynamics Non-linear cubic Schrödinger equation Reduced density matrix 

Notes

Acknowledgements

Alessandro Olgiati is Partially supported by the 2014–2017 MIUR-FIR grant “Cond-Math: Condensed Matter and Mathematical Physics”, code RBFR13WAET and by Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM). The author also warmly thanks the GSSI, for the kind hospitality and financial support during a visit in L’Aquila.

References

  1. 1.
    N. Benedikter, G. de Oliveira, B. Schlein, Quantitative derivation of the Gross-Pitaevskii equation. Comm. Pure Appl. Math. 68(8), 1399–1482 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    C. Brennecke, B. Schlein, Gross-Pitaevskii Dynamics for Bose-Einstein Condensates (2017). arXiv:1702.05625Google Scholar
  3. 3.
    L. Erdős, B. Schlein, H. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. math. 167, 515–614 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L. Erdős, B. Schlein, H. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. 172(1), 291–370 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Jeblick, N. Leopold, P. Pickl, Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions (2016). arXiv:1608.05326Google Scholar
  6. 6.
    A. Knowles, P. Pickl, Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–138 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Lührmann, Mean-field quantum dynamics with magnetic fields. J. Math. Phys. 53(2), 022105 (2012)Google Scholar
  8. 8.
    P. Pickl, A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97(2), 151–164 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation with external fields. Rev. Math. Phys. 27(1), 1550003 (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.SISSA - International School for Advanced StudiesTriesteItaly

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