Skip to main content
SpringerLink
Log in

On Effective PDEs of Quantum Physics

  • Chapter
  • First Online:
New Tools for Nonlinear PDEs and Application

Part of the book series: Trends in Mathematics ((TM))

Abstract

The Hartree-Fock equation is a key effective equation of quantum physics. We review the standard derivation of this equation and its properties and present some recent results on its natural extensions – the density functional, Bogolubov-de Gennes and Hartree-Fock-Bogolubov equations. This paper is based on a talk given at ISAAC2017.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For application of the quasifree states in the classical kinetic theory see [46].

References

  1. A. Anantharaman, E. Cancès, Existence of minimizers for Kohn-Sham models in quantum chemistry. Ann. Inst. Henri Poincarè (C) 26, 2425–2455 (2009)

    Article  MathSciNet  Google Scholar 

  2. V. Bach, E. Lieb, J.P. Solovej, Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994)

    Article  MathSciNet  Google Scholar 

  3. V. Bach, S. Breteaux, T. Chen, J. Fröhlich, I.M. Sigal, The time-dependent Hartree-Fock-Bogoliubov equations for Bosons, arXiv 2016 (https://arxiv.org/abs/1602.05171v1) and (revision) arXiv 2018 (https://arxiv.org/abs/1602.05171v2)

  4. V. Bach, S. Breteaux, T. Chen, J. Fröhlich, I.M. Sigal, On the local existence of the time-dependent Hartree-Fock-Bogoliubov equations for Bosons. arXiv 2018 (https://arxiv.org/abs/1805.04689)

  5. N. Benedikter, J. Sok, J.P. Solovej, The Dirac-Frenkel principle for reduced density matrices, and the Bogoliubov-de-Gennes equations. arXiv:1706.03082

    Google Scholar 

  6. N. Benedikter, M. Porta, B. Schlein, Hartree-Fock dynamics for weakly interacting fermions, in Mathematical Results in Quantum Mechanics (World Scientific Publishing, Hackensack, 2015), pp. 177–189

    MATH  Google Scholar 

  7. N. Benedikter, M. Porta, B. Schlein, Mean-field evolution of fermionic systems. Commun. Math. Phys. 331 1087–1131 (2014)

    Article  MathSciNet  Google Scholar 

  8. A. Bove, G. Da Prato, G. Fano, On the Hartree-Fock time-dependent problem. Commun. Math. Phys. 49, 25–33 (1976)

    Article  MathSciNet  Google Scholar 

  9. O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. 2 (Springer, Berlin, 1997)

    Book  Google Scholar 

  10. G. Bräunlich, C. Hainzl, R. Seiringer, Translation-invariant quasifree states for fermionic systems and the BCS approximation. Rev. Math. Phys. 26(7), 1450012 (2014)

    Google Scholar 

  11. E. Cancès, A. Deleurence, M. Lewin, A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case. Commun. Math. Phys. 281(1), 129–177 (2008)

    Article  Google Scholar 

  12. E. Cancès, A. Deleurence, M. Lewin, Non-perturbative embedding of local defects in crystalline materials. J. Phys. Condens. Mat. 20, 294–213 (2008)

    Article  Google Scholar 

  13. E. Cancès, M. Lewin, The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. Arch. Ration. Mech. Anal. 197, 139–177 (2010)

    Article  MathSciNet  Google Scholar 

  14. E. Cancès, M. Lewin, G. Stolz, The microscopic origin of the macroscopic dielectric permittivity of crystals: a mathematical viewpoint. arXiv 2010 (https://arxiv.org/abs/1010.3494)

  15. E. Cancès, N. Mourad, A mathematical perspective on density functional perturbation theory. Nonlinearity 27(9), 1999–2033 (2014)

    Article  MathSciNet  Google Scholar 

  16. R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations (World Scientific, Singapore, 2008)

    Book  Google Scholar 

  17. I. Catto, C. Le Bris, P.-L. Lions, On some periodic Hartree type models. Ann. Inst. H. Poincaré Anal. Non Lineaire 19(2), 143–190 (2002)

    Article  MathSciNet  Google Scholar 

  18. I. Catto, C. Le Bris, P.-L. Lions, On the thermodynamic limit for Hartree-Fock type models. Ann. Inst. H. Poincaré Anal. Non Lineaire 18(6), 687–760 (2001)

    Article  MathSciNet  Google Scholar 

  19. T. Cazenave, Semilinear Schrödinger Equations (AMS, Providence, 2003)

    Google Scholar 

  20. J.M. Chadam, The time-dependent Hartree-Fock equations with Coulomb two-body interaction. Commun. Math. Phys. 46(2), 99–104 (1976)

    Article  MathSciNet  Google Scholar 

  21. J. Chadam, R. Glassey, Global existence of solutions to the Cauchy problem for time dependent Hartree equations. J. Math. Phys. 16, 1122 (1975)

    Article  MathSciNet  Google Scholar 

  22. I. Chenn, I.M. Sigal, Stationary states of the Bogolubov-de Gennes equations. arXiv:1701.06080 (2019)

    Google Scholar 

  23. I. Chenn, I.M. Sigal, On density functional theory (2019, In preparation)

    Google Scholar 

  24. M. Cyrot, Ginzburg-Landau theory for superconductors. Rep. Prog. Phys. 36(2), 103–158 (1973)

    Article  Google Scholar 

  25. P.G. de Gennes, Superconductivity of Metals and Alloys, vol. 86 (WA Benjamin, New York, 1966)

    MATH  Google Scholar 

  26. R.J. Dodd, M. Edwards, C.W. Clark, K. Burnett, Collective excitations of Bose-Einstein-condensed gases at finite temperatures. Phys. Rev. A 57, R32–R35 (1998). https://doi.org/10.1103/PhysRevA.57.R32

    Article  Google Scholar 

  27. W. E, J. Lu, Electronic structure of smoothly deformed crystals: Cauchy-Born rule for the nonlinear tight-binding model. Commun. Pure Appl. Math. 63(11), 1432–1468 (2010)

    Google Scholar 

  28. W. E, J. Lu, The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born rule. Arch. Ration. Mech. Anal. 199(2), 407–433 (2011)

    Google Scholar 

  29. W. E, J. Lu, The Kohn-Sham Equation for Deformed Crystals (Memoirs of the American Mathematical Society, 2013), 97 pp.; Softcover MSC: Primary 74; Secondary 35

    Google Scholar 

  30. E. Elgart, L. Erdös, B. Schlein, H.-T. Yau, Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. 83(9), 1241–1273 (2004)

    Article  MathSciNet  Google Scholar 

  31. E. Elgart, L. Erdös, B. Schlein, H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Inv. Math. 167, 515–614 (2006)

    MATH  Google Scholar 

  32. E. Elgart, L. Erdös, B. Schlein, H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. (2) 172(1), 291–370 (2010)

    Google Scholar 

  33. R. Frank, C. Hainzl, R. Seiringer, J.-P. Solovej, Microscopic derivation of Ginzburg-Landau theory. J. Am. Math. Soc. 25, 667–713 (2012) and arXiv 2011 (https://arxiv.org/abs/1102.4001)

  34. M. Grillakis, M. Machedon, Pair excitations and the mean field approximation of interacting Bosons, II. arXiv 2015 (http://arxiv.org/abs/1509.05911)

  35. A. Griffin, Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures. Phys. Rev. B 53, 9341–9347 (1996). https://doi.org/10.1103/PhysRevB.53.9341

    Article  Google Scholar 

  36. S.J. Gustafson, I.M. Sigal, Mathematical Concepts of Quantum Mechanics. Universitext, 2nd edn. (Springer, Berlin, 2011)

    Google Scholar 

  37. C. Hainzl, E. Hamza, R. Seiringer, J.P. Solovej, The BCS functional for general pair interactions. Commun. Math. Phys. 281, 349–367 (2008)

    Article  MathSciNet  Google Scholar 

  38. C. Hainzl, R. Seiringer, The Bardeen-Cooper-Schrieffer functional of superconductivity and its mathematical properties. J. Math. Phys. 57, 021101 (2016)

    Article  MathSciNet  Google Scholar 

  39. C. Le Bris, P.-L. Pierre-Louis Lions, From atoms to crystals: a mathematical journey. Bull. Am. Math. Soc. 42, 291–363 (2005)

    Article  MathSciNet  Google Scholar 

  40. M. Lewin, P.T. Nam, B. Schlein, Fluctuations around Hartree states in the mean-field regime. Am. J. Math. 137(6), 1613–1650 (2015)

    Article  MathSciNet  Google Scholar 

  41. E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, The Mathematics of the Bose Gas and its Condensation. Oberwolfach Seminars Series, vol. 34 (Birkhaeuser Verlag, Basel, 2005)

    Google Scholar 

  42. E.H. Lieb, B. Simon, The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53(3), 185–194 (1977)

    Article  MathSciNet  Google Scholar 

  43. G. Lindblad, Expectations and entropy inequalities for finite quantum systems. Commun. Math. Phys. 39, 111–119 (1974)

    Article  MathSciNet  Google Scholar 

  44. P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1987)

    Article  MathSciNet  Google Scholar 

  45. P.L. Lions, Hartree-Fock and related equations. Nonlinear partial differential equations and their applications. Collège de France Seminar, vol. IX. Pitman Research Notes in Mathematics Series, vol. 181 (Pitman Advanced Publishing, Boston, 1988), pp. 304–333

    Google Scholar 

  46. J. Lukkarinen, H. Spohn, Not to normal order – notes on the kinetic limit for weakly interacting quantum fluids. J. Stat. Phys. 134, 1133–1172 (2009)

    Article  MathSciNet  Google Scholar 

  47. P.A. Markowich, G. Rein, G. Wolansky, Existence and nonlinear stability of stationary states of the Schrödinger – Poisson system. J. Stat. Phys. 106(5), 1221–1239 (2002)

    Article  Google Scholar 

  48. P.T. Nam, M. Napiórkowski, Bogoliubov correction to the mean-field dynamics of interacting bosons. [arXiv:1509.04631]

    Google Scholar 

  49. M. Napiórkowski, R. Reuvers, J.P. Solovej, The Bogoliubov free energy functional I. Existence of minimizers and phase diagram. Arch. Ration. Mech. Anal. 1–54 (2018)

    Google Scholar 

  50. M. Napiórkowski, R. Reuvers, J.P. Solovej, The Bogoliubov free energy functional II. The dilute limit. Commun. Math. Phys. 360(1), 347–403 (2018)

    Article  MathSciNet  Google Scholar 

  51. A.S. Parkins, D.F. Walls, The physics of trapped dilute-gas Bose-Einstein condensates. Phys. Rep. 303(1), 1–80 (1998)

    Article  Google Scholar 

  52. M. Porta, S. Rademacher, C. Saffirio, B. Schlein, Mean field evolution of fermions with Coulomb interaction. J. Stat. Phys. 166, 1345–1364 (2017)

    Article  MathSciNet  Google Scholar 

  53. I.M. Sigal, Magnetic vortices, Abrikosov lattices and automorphic functions, in Mathematical and Computational Modelling (With Applications in Natural and Social Sciences, Engineering, and the Arts) (Wiley, 2014)

    Google Scholar 

  54. C. Sulem, J.-P. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999)

    MATH  Google Scholar 

  55. T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis. CBMS Regional Conference Series in Mathematics, vol. 106 (AMS, Providence, 2006)

    Google Scholar 

Download references

Acknowledgements

The second author is grateful to Volker Bach, Sébastien Breteaux, Thomas Chen and Jürg Fröhlich for enjoyable collaboration, and both authors thank Dmitri Chouchkov, Rupert Frank, Christian Hainzl, Jianfeng Lu, Yuri Ovchinnikov, and especially Antoine Levitt, for stimulating discussions. The authors are grateful to the anonymous referees for useful remarks and suggestions.

The research on this paper is supported in part by NSERC Grant No. NA7901. The first author is also in part supported by NSERC CGS D graduate scholarship.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Ilias Chenn & I. M. Sigal

Authors
  1. Ilias Chenn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. M. Sigal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. M. Sigal .

Editor information

Editors and Affiliations

  1. Mathematics, University of Bari, Bari, Italy

    Marcello D'Abbicco

  2. Department of Computing and Mathematics, University of São Paulo, Ribeirão Preto, São Paulo, Brazil

    Marcelo Rempel Ebert

  3. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  4. Department of Applied Physics, Waseda University, Tokyo, Japan

    Tohru Ozawa

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chenn, I., Sigal, I.M. (2019). On Effective PDEs of Quantum Physics. In: D'Abbicco, M., Ebert, M., Georgiev, V., Ozawa, T. (eds) New Tools for Nonlinear PDEs and Application. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10937-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation