Communications in Mathematical Physics

, Volume 131, Issue 1, pp 109–124 | Cite as

Breit-Wigner formula for the scattering phase in the Stark effect

  • Markus Klein
  • Didier Robert
  • Xue-Ping Wang
Article

Abstract

Near resonance energy, we study the asymptotic behavior of the derivative of the scattering phase as the applied electric field tends to zero. We obtain the leading asymptotics of the spectral function near a simple resonance, and as an application we rigorously prove the Breit-Wigner formula which relates the width of resonances to the time delay of particles in a homogeneous electric field.

Keywords

Neural Network Statistical Physic Time Delay Complex System Asymptotic Behavior 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agmon, S., Kannai, Y.: On the asymptotic behavior of spectral functions and resolvent kernels of elliptic operators. Israël J. Math.5, 1–30 (1967)Google Scholar
  2. 2.
    Avron, J. E., Herbst, I. W.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys.52, 239–254 (1977)CrossRefGoogle Scholar
  3. 3.
    Enss, V., Simon, B.: Total cross sections in non-relativistic scattering theory. In: Quantum mechanics in mathematics, chemistry and physics. Gustafson, K. E., Reinhart, P. (eds.) New York: Plenum Press 1981Google Scholar
  4. 4.
    Gerard, C., Martinez, A.: Semiclassical asymptotics for the spectral function of long range Schrödinger Operators. J. Funct. Anal.84, 226–254 (1989)CrossRefGoogle Scholar
  5. 5.
    Gérard, C., Martinez, A., Robert, D.: Breit-Wigner formulas for the scattering phase and the total scattering cross-section in the semiclassical limit. Commun. Math. Phys.121, 323–336 (1989)CrossRefGoogle Scholar
  6. 6.
    Harrell, E., Simon, B.: The mathematical theory of resonances whose widths are exponentially small. Duke Math. J.47, 845–902 (1980)CrossRefGoogle Scholar
  7. 7.
    Helffer, B., Sjöstrand, J.: Résonances en limite semiclassique. Bull. S.M.F. Mémoire No. 24/25, tome 114 (1986)Google Scholar
  8. 8.
    Herbst, I.W.: Dilaton analytically in constant electric field. Commun. Math. Phys.64, 279–298 (1979)CrossRefGoogle Scholar
  9. 9.
    Hunziker, V.: Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poincaré45, 339–358 (1986)Google Scholar
  10. 10.
    Nakamura, S.: Scattering theory for the shape resonance model, II-Resonance scattering. Ann. Inst. H. Poincaré, (1989)Google Scholar
  11. 11.
    Newton, R. G.: Scattering theory of waves and particles, Texts and Monographs in Physics. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  12. 12.
    Robert, D., Wang, X. P.: Time-delay and spectral density for Stark Hamiltonians, I. Existence of time-delay operator. Commun. P.D.E.14, 63–98 (1989)Google Scholar
  13. 13.
    Robert, D., Wang, X. P.: Time-delay and spectral density for Stark Hamiltonians, II. Asymptotics of trace formulae (to appear)Google Scholar
  14. 14.
    Sigal, I. M.: Bounds on resonance states and width of resonances. Adv. Appl. Math. June 1988Google Scholar
  15. 15.
    Sinha, K.: Time-delay and resonances in simple scattering. In: Quantum mechanics in mathematics, chemistry and physics. Gustafson, K. E., Reinhart, P. (eds.) pp. 99–106. New York: Plenum Press 1981Google Scholar
  16. 16.
    Sjöstrand, J.: Semiclassical resonances generated by non-degenerate critical points. In: Pseudodifferential Operators. Lecture Notes in Mathematics vol.1256. pp. 402–429. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  17. 17.
    Titchmarsh, E. C.: Eigenfunction Expansions Associated with Second Order Differential Equations, II. Oxford University Press 1958Google Scholar
  18. 18.
    Wang, X. P.: Bounds on widths of resonances for Stark Hamiltonians. Acta Math. Sinica, Ser. B (1990)Google Scholar
  19. 19.
    Wang, X. P.: Asymptotics on width of resonances for Stark Hamiltonians, to appear; and also conference on “Topics on Pseudo-differential Operators”, Oberwolfach, June 1989Google Scholar
  20. 20.
    Wang, X. P.: Weak coupling asymptotics of Schrödinger operators with Stark effect, to appear. In: Harmonic Analysis. Lecture Notes in Math., Nankai subser., Berlin, Heidelberg, New York: SpringerGoogle Scholar
  21. 21.
    Yajima, K.: Spectral and scattering theory for Schrödinger operators with Stark effect, II. J. Fac. Sci. Univ. Tokyo,28A, 1–15 (1981)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Markus Klein
    • 1
  • Didier Robert
    • 2
  • Xue-Ping Wang
    • 1
    • 3
  1. 1.Fachbereich Mathematik MA 7-2Technische Universität BerlinBerlin 12FRG
  2. 2.Départment de Mathematiques et d'InformatiqueUniversité de NantesNantes CedexFrance
  3. 3.Department of MathematicsPeking UniversityBeijingChina

Personalised recommendations