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The Quantum-Mechanical Many-Body Problem: The Bose Gas

  • Conference paper
Perspectives in Analysis

Part of the book series: Mathematical Physics Studies ((MPST,volume 27))

Summary

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago, and to explore new regimes not treated before. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous — and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πħ2 aρ/m where a is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N 7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. For this system it is also possible to prove complete Bose condensation and superfluidity as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. Another topic is a proof that Foldy’s 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy; using this we can also prove the N 7/5 formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. All of this is quite recent work and it is hoped that the mathematical methodology might be useful, ultimately, to solve more complex problems connected with these interesting systems.

On leave from Dept. of Math., University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

The work was supported in part by US NSF grants PHY 0139984-A01 (EHL), PHY 0353181 (RS) and DMS-0111298 (JPS); by EU grant HPRN-CT-2002-00277 (JPS and JY); by FWF grant P17176-N02 (JY); by MaPhySto — A Network in Mathematical Physics and Stochastics funded by The Danish National Research Foundation (JPS), and by grants from the Danish research council (JPS).

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References

  1. G.E. Astrakharchik, S. Giorgini: Phys. Rev. A 66 053614-1–6 (2002)

    Google Scholar 

  2. G.E. Astrakharchik, J. Boronat, J. Casulleras et al: Phys. Rev. A 66 023603 (2002)

    Article  Google Scholar 

  3. B. Baumgartner: J. Phys. A 30, L741 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Baym. In: Math. Methods in Solid State and Superfluid Theory, Scottish Univ. Summer School of Physics (Oliver and Boyd, Edinburgh 1969)

    Google Scholar 

  5. F.A. Berezin: Izv. Akad. Nauk, ser. mat. 36 (No. 5) (1972); English translation: USSR Izv. 6 (No. 5) (1972); Commun. Math. Phys. 40, 153 (1975)

    Google Scholar 

  6. D. Blume: Phys. Rev. A 66 053613-1–8 (2002)

    Google Scholar 

  7. N.N. Bogolubov: J. Phys. (U.S.S.R.) 11, 23 (1947); N.N. Bogolubov, D.N. Zubarev: Sov. Phys.-JETP 1, 83 (1955)

    MathSciNet  Google Scholar 

  8. K. Bongs, S. Burger, S. Dettmer et al: Phys. Rev. A, 63 031602 (2001)

    Article  Google Scholar 

  9. S.N. Bose: Z. Phys. 26, 178 (1924)

    Article  MATH  Google Scholar 

  10. A.Y. Cherny, A.A. Shanenko: Phys. Rev. E 64 027105 (2001)

    Article  Google Scholar 

  11. A.Y. Cherny, A.A. Shanenko: Phys. Lett. A 293, 287 (2002)

    Article  MATH  Google Scholar 

  12. J. Conlon, E.H. Lieb, H.-T. Yau: Commun. Math. Phys. 116, 417 (1988)

    Article  MathSciNet  Google Scholar 

  13. S.L. Cornish, N.R. Claussen, J.L. Roberts et al: Phys. Rev. Lett. 85, 1795 (2000)

    Article  Google Scholar 

  14. F. Dalfovo, S. Giorgini, L.P. Pitaevskii et al: Rev. Mod. Phys. 71, 463 (1999)

    Article  Google Scholar 

  15. K.K. Das, M.D. Girardeau, E.M. Wright: Phys. Rev. Lett. 89, 110402-1–4 (2002)

    Google Scholar 

  16. V. Dunjko, V. Lorent, M. Olshanii: Phys. Rev. Lett. 86, 5413 (2001)

    Article  Google Scholar 

  17. F.J. Dyson: Phys. Rev. 106, 20 (1957)

    Article  MATH  Google Scholar 

  18. F.J. Dyson: J. Math. Phys. 8, 1538 (1967)

    Article  MathSciNet  Google Scholar 

  19. F.J. Dyson, E.H. Lieb, B. Simon: J. Stat. Phys. 18, 335 (1978)

    Article  MathSciNet  Google Scholar 

  20. A. Einstein: Sitzber. Kgl. Preuss. Akad. Wiss., 261 (1924), 3 (1925)

    Google Scholar 

  21. A.L. Fetter, A.A. Svidzinsky: J. Phys.: Condens. Matter 13, R135 (2001)

    Article  Google Scholar 

  22. D.S. Fisher, P.C. Hohenberg: Phys. Rev. B 37, 4936 (1988)

    Article  Google Scholar 

  23. L.L. Foldy: Phys. Rev. 124, 649 (1961). Errata ibid 125, 2208 (1962)

    Article  Google Scholar 

  24. M.D. Girardeau: J. Math. Phys. 1, 516 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  25. M.D. Girardeau, E.M. Wright: Phys. Rev. Lett. 87, 210402-1–4 (2001)

    Google Scholar 

  26. M.D. Girardeau, E.M. Wright, J.M. Triscari: Phys. Rev. A 63, 033601-1–6 (2001)

    Article  Google Scholar 

  27. A. Görlitz et al: Phys. Rev. Lett. 87, 130402-1–4 (2001)

    Article  Google Scholar 

  28. G.M. Graf and J.P. Solovej: Rev. Math. Phys. 6, 977 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Greiner et al: Phys. Rev. Lett. 87, 160405 (2001)

    Article  Google Scholar 

  30. E.P. Gross: Nuovo Cimento 20, 454 (1961)

    MATH  Google Scholar 

  31. E.P. Gross: J. Math. Phys. 4, 195 (1963)

    Article  Google Scholar 

  32. D.F. Hines, N.E. Frankel, D.J. Mitchell: Phys. Lett. 68A, 12 (1978)

    Google Scholar 

  33. P.C. Hohenberg: Phys. Rev. 158, 383 (1966)

    Article  Google Scholar 

  34. P.C. Hohenberg, P.C. Martin: Ann. Phys. (NY) 34, 291 (1965)

    Article  Google Scholar 

  35. K. Huang. In: Bose-Einstein Condensation, ed by A. Griffin, D.W. Stroke, S. Stringari (Cambridge University Press, Cambridge 1995) pp 31–50

    Google Scholar 

  36. K. Huang, C.N. Yang: Phys. Rev. 105, 767 (1957); T.D. Lee, K. Huang, C.N. Yang: Phys. Rev. 106, 1135 (1957); K.A. Brueckner, K. Sawada: Phys. Rev. 106, 1117, 1128 (1957); S.T. Beliaev: Sov. Phys.-JETP 7, 299 (1958); T.T. Wu: Phys. Rev. 115, 1390 (1959); N. Hugenholtz, D. Pines: Phys. Rev. 116, 489 (1959); M. Girardeau, R. Arnowitt: Phys. Rev. 113, 755 (1959); T.D. Lee, C.N. Yang: Phys. Rev. 117, 12 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  37. A.D. Jackson, G.M. Kavoulakis: Phys. Rev. Lett. 89, 070403 (2002)

    Article  Google Scholar 

  38. T. Kennedy, E.H. Lieb, S. Shastry: Phys. Rev. Lett. 61, 2582 (1988)

    Article  Google Scholar 

  39. W. Ketterle, N.J. van Druten: Evaporative Cooling of Trapped Atoms. In: Advances in Atomic, Molecular and Optical Physics, vol 37, ed by B. Bederson, H. Walther (Academic Press 1996) pp 181–236

    Google Scholar 

  40. E.B. Kolomeisky, T.J. Newman, J.P. Straley et al: Phys. Rev. Lett. 85, 1146 (2000)

    Article  Google Scholar 

  41. M. Kobayashi, M. Tsubota: Phys. Rev. B 66, 174516 (2002)

    Article  Google Scholar 

  42. S. Komineas, N. Papanicolaou: Phys. Rev. Lett. 89, 070402 (2002)

    Article  Google Scholar 

  43. A. Lenard: J. Math. Phys. 5, 930 (1964)

    Article  MathSciNet  Google Scholar 

  44. E.H. Lieb: Phys. Rev. 130, 2518 (1963). See also Phys. Rev. 133 A899 (1964), (with A.Y. Sakakura) and Phys. Rev. 134 A312 (1964), (with W. Liniger)

    Article  Google Scholar 

  45. E.H. Lieb: The Bose fluid. In: Lecture Notes in Theoretical Physics vol VIIC, ed by W.E. Brittin (Univ. of Colorado Press 1964) pp 175–224

    Google Scholar 

  46. E.H. Lieb: Commun. Math. Phys. 31, 327 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  47. E.H. Lieb: The Bose Gas: A Subtle Many-Body Problem. In Proceedings of the XIII International Congress on Mathematical Physics, London, ed by A. Fokas, et al. (International Press 2001) pp 91–111

    Google Scholar 

  48. E.H. Lieb, W. Liniger: Phys. Rev. 130, 1605 (1963); E.H. Lieb: Phys. Rev. 130, 1616 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  49. E.H. Lieb, M. Loss: Analysis, 2nd edn (Amer. Math. Society, Providence, R.I. 2001)

    MATH  Google Scholar 

  50. E.H. Lieb, H. Narnhofer: J. Stat. Phys. 12, 291 (1975); Errata J. Stat. Phys. 14, 465 (1976)

    Article  MathSciNet  Google Scholar 

  51. E.H. Lieb, R. Seiringer: Phys. Rev. Lett. 88, 170409-1–4 (2002)

    Article  Google Scholar 

  52. E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason: The ground state of the Bose gas. In: Current Developments in Mathematics (International Press, Cambridge 2002) pp 131–178

    Google Scholar 

  53. E.H. Lieb, R. Seiringer, J. Yngvason: Phys. Rev A 61, 043602 (2000)

    Article  Google Scholar 

  54. E.H. Lieb, R. Seiringer, J. Yngvason: Commun. Math. Phys. 224, 17 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  55. E.H. Lieb, R. Seiringer, J. Yngvason: The Ground State Energy and Density of Interacting Bosons in a Trap. In: Quantum Theory and Symmetries, ed by H.-D. Doebner, V.K. Dobrev, J.-D. Hennig, W. Luecke (World Scientific 2000) pp 101–110

    Google Scholar 

  56. E.H. Lieb, R. Seiringer, J. Yngvason: Two-Dimensional Gross-Pitaevskii Theory. In: Progress in Nonlinear Science, Proceedings of the International Conference Dedicated to the 100th Anniversary of A.A. Andronov, vol II, ed by A.G. Litvak (Nizhny Novgorod, Institute of Applied Physics, University of Nizhny Novgorod 2002) pp 582–590

    Google Scholar 

  57. E.H. Lieb, R. Seiringer, J. Yngvason: Phys. Rev. B 66, 134529 (2002)

    Article  Google Scholar 

  58. E.H. Lieb, R. Seiringer, J. Yngvason: Commun. Math. Phys. 244, 347 (2004); See also: Phys. Rev. Lett. 91, 150401-1-4 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  59. E.H. Lieb, R. Seiringer, J. Yngvason: Ann. Math. 158, 1067 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  60. E.H. Lieb, J.P. Solovej: Commun. Math. Phys. 217, 127 (2001); Errata 225, 219 (2002)

    Article  MathSciNet  Google Scholar 

  61. E.H. Lieb, J.P. Solovej: Commun. Math. Phys. 252, 485 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  62. E.H. Lieb, J. Yngvason: Phys. Rev. Lett. 80, 2504 (1998)

    Article  Google Scholar 

  63. E.H. Lieb, J. Yngvason: J. Stat. Phys. 103, 509 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  64. E.H. Lieb, J. Yngvason: The Ground State Energy of a Dilute Bose Gas. In: Differential Equations and Mathematical Physics, ed by R. Weikard, G. Weinstein (University of Alabama, Birmingham 1999) pp 271–282

    Google Scholar 

  65. H. Moritz, T. Stöferle, M. Köhl et al: Phys. Rev. Lett. 91, 250402 (2003)

    Article  Google Scholar 

  66. W.J. Mullin: J. Low Temp. Phys. 106, 615 (1997)

    Article  Google Scholar 

  67. M. Olshanii: Phys. Rev. Lett. 81, 938 (1998)

    Article  Google Scholar 

  68. A.A. Ovchinnikov: J. Phys. Condens. Matter 5, 8665 (1993); See also JETP Letters 57, 477 (1993); Mod. Phys. Lett. 7, 1029 (1993)

    Article  Google Scholar 

  69. C. Pethick, H. Smith: Bose Einstein Condensation of Dilute Gases, (Cambridge University Press 2001)

    Google Scholar 

  70. D.S. Petrov, G.V. Shlyapnikov, J.T.M. Walraven: Phys. Rev. Lett. 85, 3745 (2000)

    Article  Google Scholar 

  71. L.P. Pitaevskii: Sov. Phys. JETP. 13, 451 (1961)

    MathSciNet  Google Scholar 

  72. L. Pitaevskii, S. Stringari: J. Low Temp. Phys. 85, 377 (1991)

    Article  Google Scholar 

  73. V.N. Popov: Theor. and Math. Phys. 11, 565 (1977)

    Article  Google Scholar 

  74. N.V. Prokof’ev, B.V. Svistunov: Phys. Rev. B 61, 11282 (2000)

    Article  Google Scholar 

  75. M. Schick: Phys. Rev. A 3, 1067 (1971)

    Article  Google Scholar 

  76. F. Schreck et al: Phys. Rev. Lett. 87, 080403 (2001)

    Article  Google Scholar 

  77. R. Seiringer: Diplom thesis, University of Vienna, (1999)

    Google Scholar 

  78. R. Seiringer: Bosons in a Trap: Asymptotic Exactness of the Gross—Pitaevskii Ground State Energy Formula. In: Partial Differential Equations and Spectral Theory, PDE2000 Conference in Clausthal, Germany, ed by M. Demuth, B.-W. Schulze (Birkhäuser 2001) pp 307–314

    Google Scholar 

  79. R. Seiringer: Commun. Math. Phys. 229, 491 (2002); J. Phys. A: Math. Gen. 36, 9755 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  80. S.I. Shevchenko: Sov. J. Low Temp. Phys. 18, 223 (1992)

    Google Scholar 

  81. B. Simon: Trace ideals and their application (Cambridge University Press 1979)

    Google Scholar 

  82. J.P. Solovej: Upper Bounds to the Ground State Energies of the One-and Two-Component Charged Bose gases, preprint, arxiv:math-ph/0406014

    Google Scholar 

  83. G. Temple: Proc. Roy. Soc. London A 119, 276 (1928)

    MATH  Google Scholar 

  84. D.R. Tilley, J. Tilley: Superfluidity and Superconductivity, 3rd edn (Adam Hilger, Bristol New York 1990)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, Jadwin Hall, P.O. Box 708, Princeton, NJ, 08544, USA

    Elliott H. Lieb

  2. Department of Physics, Princeton University, Jadwin Hall, P.O. Box 708, Princeton, NJ, 08544, USA

    Robert Seiringer

  3. School of Mathematics, Institute for Advanced Study, 1 Einstein Dr., Princeton, N.J., 08540

    Jan Philip Solovej

  4. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090, Vienna, Austria

    Jakob Yngvason

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  1. Elliott H. Lieb
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  2. Robert Seiringer
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  3. Jan Philip Solovej
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  4. Jakob Yngvason
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Michael Benedicks  & Stanislav Smirnov  & 

  2. Department of Mathematics, Yale University, 10 Hillhouse Ave, P.O. Box 208283, New Haven, CT, 06520-8283, USA

    Peter W. Jones

  3. Section de Mathématiques, Université de Genève, 2-4, rue du Lièvre, 1211, Genève 4, Switzerland

    Stanislav Smirnov

  4. Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    Björn Winckler

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Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J. (2005). The Quantum-Mechanical Many-Body Problem: The Bose Gas. In: Benedicks, M., Jones, P.W., Smirnov, S., Winckler, B. (eds) Perspectives in Analysis. Mathematical Physics Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-30434-7_9

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