Il Nuovo Cimento B (1965-1970)

, Volume 56, Issue 1, pp 71–82 | Cite as

Lower bounds to the ground-state energy ofN-body systems and conditions on the two-body potentials necessary for the existence ofN-body bound states

  • F. Calogero
  • Yu. A. Simonov
Article
First Online:
Received:

Summary

We obtain simple lower bounds to the ground-state energy ofN-body systems, and conditions on the two-body forces which are necessary for the existence ofN-body bound states. The simpler version of these conditions reads (with an obvious meaning of the symbols)\( - 2h / - 2\int\limits_0^\infty {drr\sum\limits_{i > j}^N {m_i m_j } (m_i + m_j )^{ - 1} V_{ij} (r)\theta [ - V_{ij} (r)] \geqslant 1} \) thus it provides the directN-body generalization of the well-known Jost-Pais condition.

Keywords

Green Function Trial Function Obvious Meaning Assigned Quantum Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Нижние границы для знергии основного состояния системN-частиц и условия на двух-частичные потенциалы, необходимые для сушествования свяэанных состоянийN-частиц

Реэюме

Мы получаем простые нижние границы для знергии основного состояния системN-частиц и условия на двух-частичные силы, которые необходимы для сушествования свяэанных состоянийN-частиц. Простейщий вариант зтих условий имеет вид (эначение символов очевидно)\( - 2h / - 2\int\limits_0^\infty {drr\sum\limits_{i > j}^N {m_i m_j } (m_i + m_j )^{ - 1} V_{ij} (r)\theta [ - V_{ij} (r)] \geqslant 1} \); что представляет непосредственноеN-частичное обобшение хорощо иэвестного условия йста-Пейса.

Riassunto

Si ottengono semplici limiti inferiori per l’energia del livello fondamentale di un sistema diN corpi, e condizioni sulle forze a due corpi che sono necessarie per l’esistenza di stati legati aN corpi. La più semplice di tali condizioni recita (con ovvio significato dei simboli) —\( - 2h / - 2\int\limits_0^\infty {drr\sum\limits_{i > j}^N {m_i m_j } (m_i + m_j )^{ - 1} V_{ij} (r)\theta [ - V_{ij} (r)] \geqslant 1} \), sicché costituisce una diretta generalizzazione al caso diN corpi della ben nota condizione di Jost e Pais.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    F. Calogero andYu. A. Simonov:Phys. Rev. (in press).Google Scholar
  2. (2).
    L. Spruch:Minimum Principles in Scattering Theory, inLectures in Theoretical Physics, Boulder, Colo., 1961, edited byE. W. Brittin, W. B. Downs andJ. Downs (New York, 1962).Google Scholar
  3. (3).
    L. Spruch:Necessary conditions for the existence of bound states, inFew-Nucleon Problems, Proceedings of the Ninth Summer Meeting of Nuclear Physicists, Hercegnovi, 1964, edited byM. Cerineo, published by the Federal Nuclear Energy Commission of Yugoslavia, vol.1, p. 271. The possibility that this approach had been previously used by Svartholm in his thesis at Lund in 1945 is mentioned by Spruch (see p. 309).Google Scholar
  4. (4).
    R. Jost andA. Pais:Phys. Rev.,82, 840 (1951). See also:V. Bargmann:Proc. Acad. Sci. U.S.A.,38, 961 (1952).ADSMathSciNetCrossRefGoogle Scholar
  5. (5).
    J. Schwinger:Proc. Acad. Sci. U.S.A.,47, 122 (1961).ADSMathSciNetCrossRefGoogle Scholar
  6. (6).
    G. C. Ghirardi andA. Rimini:Nuovo Cimento,37, 450 (1965).MathSciNetCrossRefGoogle Scholar
  7. (7).
    L. Fonda andG. C. Ghirardi:Nuovo Cimento,46, 47 (1966).ADSCrossRefGoogle Scholar
  8. (8).
    L. D. Faddeev:Žurn. Ėksp. Teor. Fiz.,39, 1459 (1960) (English translation:Sov. Phys. JETP,12, 1014 (1961));Dokl. Acad. Nauk. SSSR,138, 565 (1961);145, 301 (1962) (Sov. Phys. Dokl. 6, 384 (1961);7, 600 (1963));C. Lovelace:Phys. Rev.,135, B 1225 (1964).MathSciNetGoogle Scholar
  9. (9).
    We also require ∥Ψ(r)/Ψ t(r)∥ to be finite; this is certainly true ifΦ t(>r) is positive definite.Google Scholar
  10. (10).
    In fact, as emphasized bySpruch (3), one advantage of this approach is the fact that it yields results even for noncentral potentials. The form of these results is obvious and is not given explicitly.Google Scholar
  11. (11).
    In fact, with this choice of the trial functionΦ t(r), the equality sign prevails in eq. (2.9). This choice is allowed unless the two particles are identical fermions, becauseΦ t(r) is by assumption positive semi-definite.Google Scholar
  12. (14).
    The choice of a trial function such as that used in ref.(1) is appealing, but we do not discuss this possibility in this paper.Google Scholar
  13. (15).
    In this and in the following inequalities, the equality sign should be omitted forN>2; in fact it is easily seen that, in contrast to the two-body case, in theN-body case (withN>2) the choiceΦ t= const cannot reproduce the exact result.Google Scholar
  14. (16).
    S. Weinberg:Phys. Rev.,133, B 232 (1964).Google Scholar
  15. (17).
    E. Erdélyi (ed.):Higher Transcendental Functions, vol.2 (New York, 1953), eq. (7.12) (23).Google Scholar
  16. (18).
    A. M. Badalyan andYu. A. Simonov:Journ. Nucl. Phys.,3, 1032 (1966) (English translation:Sov. JNP,3, 755 (1966)).Google Scholar
  17. (19).
    Ref. (17), p. 62.Google Scholar
  18. (20).
    Ref. (17), eq. (7.13) (7).Google Scholar

Copyright information

© Società Italiana di Fisica 1968

Authors and Affiliations

  • F. Calogero
    • 1
    • 2
  • Yu. A. Simonov
    • 3
  1. 1.Istituto di Fisica dell’UniversitàRoma
  2. 2.Istituto Nazionale di Fisica Nucleare — Sezione di RomaItaly
  3. 3.Institute of Theoretical and Experimental PhysicsMoscow

Personalised recommendations