A remark on bound states in potential-scattering theory (original) (raw)

Summary

Let ℋ = ℋ B + ℋ C be the Hilbert space of an_n_-particle quantum system, where ℋ B is spanned by the bound states and ℋ C corresponds to the continuous spectrum of the Hamiltonian. It is shown that the wave functions which are in some sense localized in space and energy form a compact set in ℋ. This is used to prove that a wave packet ψ remains localized at finite distance for all time if ψ∈ℋ B , and that it disappears at infinity if ψ∈ℋ C .

Riassunto

Sia ℋ = ℋ B + ℋ C lo spazio hilbertiano di un sistema quantistico di_n_ particelle, in cui ℋ B è coperto dagli stati legati e ℋ C corrisponde allo spettro continuo dell'hamiltoniana. Si dimostra che le funzioni d'onda che sono in un certo senso localizzate nello spazio e nell'energia formano un insieme compatto in ℋ. Da ciò si dimostra che un pacchetto d'onde ψ rimane localizzato ad una distanza finita in tutti gli istanti se ψ∈ℋ B , e che scompare all'infinito se ψ∈ℋ C .

Access this article

Log in via an institution

Subscribe and save

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

Literatur

  1. R. Haag andJ. A. Swieca:Comm. Math. Phys.,1, 308 (1965).
    Article ADS MathSciNet Google Scholar
  2. F. Riesz andB. Sz.-Nagy:Leçons d'Analyse Fonctionelle, Académie des Sciences de Hongrie, 1955.
  3. T. Kato:Trans. Am. Math. Soc.,70, 195 (1951).
    Google Scholar
  4. T. Kato:Perturbation Theory of Linear Operators (Berlin, 1966).
  5. E. Nelson:Journ. Math. Phys.,5, 332 (1964).
    Article ADS Google Scholar
  6. W. Hunziker:Helv. Phys. Acta,39, 451 (1966).
    MathSciNet Google Scholar
  7. K. Jacobs:Lecture Notes on Ergodic Theory, Aarhus Universitet, Aarhus, 1963.
    MATH Google Scholar

Download references

Author information

Author notes

  1. D. Ruelle
    Present address: IHES, 91 Bures-sur-Yvette

Authors and Affiliations

  1. University of California, Irvine
    D. Ruelle

Authors

  1. D. Ruelle
    You can also search for this author inPubMed Google Scholar

Additional information

Traduzione a cura della Redazione.

Rights and permissions

About this article

Cite this article

Ruelle, D. A remark on bound states in potential-scattering theory.Nuovo Cimento A (1965-1970) 61, 655–662 (1969). https://doi.org/10.1007/BF02819607

Download citation

Keywords