Skip to main content
SpringerLink
Log in

Quantum mechanical low energy scattering in terms of diffusion processes

  • Conference paper
  • First Online:
Stochastic Aspects of Classical and Quantum Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1109))

Abstract

We give expressions for the scattering lengths as expectations with respect to the stochastic process associated with the quantum mechanical Hamiltonian and the corresponding diffusion Dirichlet operator. We also briefly discuss the asymptotics of the drift in relation to the scattering length.

Alexander von Humboldt Research Fellow

On leave of absence from Institut für Theoretische Physik, Universität Graz, Austria

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 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Simon, "Quantum Mechanics for Hamiltonians defined as Quadratic Forms", Princeton Univ. Press, 1971

    Google Scholar 

  2. M. Reed and B. Simon, "Methods of Modern Mathematical Physics Vol. III: Scattering Theory", Academic Press, New York, 1979

    MATH  Google Scholar 

  3. A. Grossmann and T.T. Wu, J. Math. Phys. 2, 710 (1961)

    Article  Google Scholar 

  4. A. Grossmann, J. Math. Phys. 2, 714 (1961)

    Article  Google Scholar 

  5. A. Grossmann and T.T. Wu, J. Math. Phys. 3, 684 (1962)

    Article  Google Scholar 

  6. S. Albeverio and R. Høegh-Krohn, J. Operator Theory 6, 313 (1980)

    Google Scholar 

  7. S. Albeverio, F. Gesztesy and R. Høegh-Krohn, Ann. Inst. H. Poincaré A37, 1 (1982

    Google Scholar 

  8. A. Jensen and T. Kato, Duke Math. J. 46, 583 (1979)

    Article  MathSciNet  Google Scholar 

  9. M. Klaus and B. Simon, Ann. Phys. 130, 251 (1980)

    Article  MathSciNet  Google Scholar 

  10. R.G. Newton, "Scattering Theory of Waves and Particles", 2nd ed., Springer, New York, 1982

    Book  MATH  Google Scholar 

  11. R.G. Newton, J. Math. Phys. 18, 1348 (1977)

    Article  Google Scholar 

  12. S. Albeverio, D. Bollé, F. Gesztesy, R. Høegh-Krohn and L. Streit, Ann. Phys. 148, 308 (1983)

    Article  Google Scholar 

  13. B. Simon, "Functional Integration and Quantum Physics", Academic Press, New York, 1979

    MATH  Google Scholar 

  14. B. Simon, J. Func. Anal. 35, 215 (1980)

    Article  Google Scholar 

  15. B. Simon, J. Func. Anal. 40, 66 (1981)

    Article  Google Scholar 

  16. B. Simon, Bull. Am. Math. Soc. 7, 447 (1982)

    Article  Google Scholar 

  17. M. Aizenman and B. Simon, Commun. Pure Appl. Math. 35, 209 (1982)

    Article  MathSciNet  Google Scholar 

  18. D.V. Widder, "The Laplace Transform", Princeton Univ. Press, 1946

    Google Scholar 

  19. R. Carmona, J. Func. Anal. 33, 259 (1979)

    Article  MathSciNet  Google Scholar 

  20. M. Kac, Rocky Mountain J. Math. 4, 511 (1974)

    Article  MathSciNet  Google Scholar 

  21. M. Kac and J.M. Luttinger, Ann. Inst. Fourier 25, 317 (1975)

    Article  MathSciNet  Google Scholar 

  22. M.E. Taylor, J. Math. Anal. Appl. 53, 291 (1976)

    Article  MathSciNet  Google Scholar 

  23. T. Ikebe, Arch. Rat. Mech. Anal. 5, 1 (1960)

    Article  MathSciNet  Google Scholar 

  24. G.A. Hagedorn and P.A. Perry, Commun. Pure Appl. Math. 36, 213 (1983)

    Article  MathSciNet  Google Scholar 

  25. M. Fukushima, "Dirichlet Forms and Markov Processes", North-Holland/Kodansha, Amsterdam, 1980

    MATH  Google Scholar 

  26. S. Albeverio, R. Høegh-Krohn and L. Streit, J. Math. Phys. 18, 907 (1977)

    Article  Google Scholar 

  27. S. Albeverio, R. Høegh-Krohn and L. Streit, J. Math. Phys. 21, 1636 (1980)

    Article  MathSciNet  Google Scholar 

  28. M. Fukushima, Phys. Rep. 77, 255 (1981)

    Article  MathSciNet  Google Scholar 

  29. M. Fukushima, "On a representation of local martingale additive functionals of symmetric diffusions", pp. 110–118 in "Stochastic Integrals", Proc., LMS Durham Symp. 1980, D. Williams (ed.), Lecture Notes in Math. 851, Springer, Berlin, 1981

    Google Scholar 

  30. S. Albeverio, F. Gesztesy, W. Karwowski and L. Streit, "On the connection between Schrödinger and Dirichlet forms", ZiF-Preprint, Univ. Bielefeld, 1984

    Google Scholar 

  31. P. Biler, Lett. Math. Phys. 8, 1 (1984)

    Article  MathSciNet  Google Scholar 

  32. M. Röckner, Applic. Math., 1984

    Google Scholar 

  33. C. DeWitt-Morette and K.D. Elworthy, New stochastic methods in physics, Edts., Phys. Reports 77, No. 3 (1981)

    Google Scholar 

  34. E. Nelson, Dynamical theories of Brownian motion, Princeton Univ. Press, 1966

    Google Scholar 

  35. E. Nelson, Quantum fluctuations, book in preparation

    Google Scholar 

  36. F. Guerra, Phys. Repts. 77, 263–312 (1981)

    Article  MathSciNet  Google Scholar 

  37. J u.G. Kondratsev and V.D. Koshmanenko, Sov. Math. Doklady 26, 585–589 (1982)

    Google Scholar 

  38. S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, Non Standard methods in stochastic analysis and mathematical physics, Acad. Press 1984

    Google Scholar 

  39. E. Carlen, Conservative diffusions and constructive approach to Nelson stochastic mechanics, Diss. Princeton Univ., June 84.

    Google Scholar 

  40. S. Albeverio, R. Høegh-Krohn, J. Math. Phys. 15, 1745–1747 (1974).

    Article  Google Scholar 

  41. L. Streit, Phys. Repts. 77, 363–375 (1981)

    Article  MathSciNet  Google Scholar 

  42. G. Jona-Lasinio, F. Martinelli, E. Scoppola, Comm. Math. Phys. 80, 223–254 (1981).

    Article  MathSciNet  Google Scholar 

  43. S. Albeverio, T. Arede, The relation between quantum mechanics and classical mechanics: a survey of mathematical aspects, Proc. Como 83 Conf. Quantum Chaos, Plenum

    Google Scholar 

  44. S. Albeverio, R. Høegh-Krohn, pp. 497–540 in Ref. [29]

    Google Scholar 

  45. S. Albeverio, R. Høegh-Krohn, Diffusion fields, quantum fields, and fields with values in Lie groups, in Stochastic Analysis and Applications, Ed. M.A. Pinsky, M. Dekker, 1984.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Ruhr Universität, D-4630, Bochum, Germany

    S. Albeverio

  2. Theoretische Physik, Universität Bielefeld, D-4800, Bielefeld, Germany

    Ph. Blanchard

  3. Zentrum für interdisziplinäre Forschung, Universität Bielefeld, D-4800, Bielefeld, Germany

    F. Gesztesy

  4. Institut für Theoretische Physik, Universität Bielefeld, D-4800, Bielefeld

    L. Streit

Authors
  1. S. Albeverio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ph. Blanchard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Gesztesy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. Streit
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Sergio Albeverio Philippe Combe Madeleine Sirugue-Collin

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Albeverio, S., Blanchard, P., Gesztesy, F., Streit, L. (1985). Quantum mechanical low energy scattering in terms of diffusion processes. In: Albeverio, S., Combe, P., Sirugue-Collin, M. (eds) Stochastic Aspects of Classical and Quantum Systems. Lecture Notes in Mathematics, vol 1109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101546

Download citation

  • DOI: https://doi.org/10.1007/BFb0101546

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13914-0

  • Online ISBN: 978-3-540-39138-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation