Skip to main content
SpringerLink
Log in

Weak asymptotics of the wave function for an \(N\)-particle system and asymptotic filtration

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We construct asymptotic representations for large values of the hyperradius for the scattering wave function of an \(N\)-particles system regarded as a generalized function of angular coordinates. We express the coefficients of the asymptotic representations in terms of the \(N\)-particle scattering matrix. We discover the phenomenon of asymptotic filtration: only scattering processes in which all particles are free both before and after interaction contribute to the leading terms of such an asymptotic representation. We use the obtained representations to construct the correct asymptotic forms of the partial components of the \(N\)-particle wave function in the hyperspherical representation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In fact, the product \(|P||X|\) is the large parameter in calculating the asymptotic behavior of wave functions, but in what follows, we assume that the value of \(|P|\) is bounded and nonzero, and requiring \(|X|\to\infty\) therefore suffices.

References

  1. S. P. Merkur’ev, “Coordinate asymptotic behavior of three-particle wave functions,” Theor. Math. Phys., 8, 798–809 (1971).

    Article  Google Scholar 

  2. S. L. Yakovlev, “Asymptotic behavior of the wave function of three particles in a continuum,” Theor. Math. Phys., 186, 126–135 (2016).

    Article  MathSciNet  Google Scholar 

  3. S. L. Yakovlev, “Coordinate asymptotics of the wave function for a system of four particles free in the initial state,” Theor. Math. Phys., 82, 157–169 (1990).

    Article  MathSciNet  Google Scholar 

  4. S. P. Merkuriev and L. D. Faddeev, Quantum Scattering Theory for Systems of Several Particles [in Russian], Nauka, Moscow (1974); English transl.: L. D. Faddeev and S. P. Merkuriev, Quantum Scattering Theory for Several Particle Systems (Math. Phys. Appl. Math., Vol. 11), Springer, Dordrecht (1993).

    Google Scholar 

  5. J. Nuttal, “Asymptotic form of the three-particle scattering wave function for free incident particles,” J. Math. Phys., 12, 1896–1899 (1971).

    Article  ADS  Google Scholar 

  6. R. G. Newton, “The asymptotic form of three-particle wavefunctions and the cross sections,” Ann. Phys., 74, 324–351 (1972).

    Article  ADS  MathSciNet  Google Scholar 

  7. S. P. Merkur’ev and S. L. Yakovlev, “Quantum \(N\)-body scattering theory in configuration space,” Theor. Math. Phys., 56, 673–682 (1983).

    Article  Google Scholar 

  8. S. L. Yakovlev, “Quantum \(N\)-body problem: Matrix structures and equations,” Theor. Math. Phys., 181, 1317–1338 (2014).

    Article  MathSciNet  Google Scholar 

  9. L. D. Faddeev, Mathematical Questions in the Quantum Theory of Scattering for a System of Three Particles (Trudy Mat. Inst. Steklov., Vol. 69), Acad. Sci. USSR, Moscow (1963).

    Google Scholar 

  10. N. Ja. Vilenkin, Special Functions and the Theory of Group Representations [in Russian], Nauka, Moscow (1965); English transl. (Transl. Math. Monogr., Vol. 22), Amer. Math. Soc., Providence, R. I. (1968).

    MATH  Google Scholar 

  11. R. I. Jibuti and N. B. Krupennikova, Hyperspherical Harmonics Method in the Quantum Mechanics of Several Bodies [in Russian], Metsniereba, Tbilisi (1984).

    Google Scholar 

Download references

Funding

This research is supported by the Russian Foundation for Basic Research (Grant No. 18-02-00492_a).

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    S. L. Yakovlev

Authors
  1. S. L. Yakovlev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. L. Yakovlev.

Ethics declarations

The author declares no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yakovlev, S.L. Weak asymptotics of the wave function for an \(N\)-particle system and asymptotic filtration. Theor Math Phys 206, 68–83 (2021). https://doi.org/10.1134/S0040577921010049

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577921010049

Keywords

Search

Navigation