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International Journal of Theoretical Physics

, Volume 46, Issue 11, pp 2688–2707 | Cite as

On a Semiclassical Formula for Non-Diagonal Matrix Elements

  • O. Lev
  • P. Šťovíček
Article
  • 40 Downloads

Abstract

Let H()=− 2d2/dx 2+V(x) be a Schrödinger operator on the real line, W(x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V(x) in exactly two turning points and lies below V =lim inf |x|→∞ V(x). We consider the semiclassical limit n→∞, = n →0 and E n =E where E n is the nth eigenenergy of H(). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.

Keywords

Semiclassical limit Non-diagonal matrix elements WKB method 

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References

  1. 1.
    Bellissard, J.: In: Albeverio, Blanchard (eds.) Trends and Developments in the Eighties, pp. 1–106. Word Scientific, Singapore (1985). Google Scholar
  2. 2.
    Bellissard, J., Vittot, M.: Ann. Inst. H. Poincaré 52, 175 (1990) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Moscow University Press, Moscow (1983) (in Russian). English transl. Kluwer, Dordrecht, 1991 zbMATHGoogle Scholar
  4. 4.
    Brummelhuis, R., Uribe, A.: Commun. Math. Phys. 136, 567 (1991) zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Charbonnel, A.-M.: Ann. Inst. H. Poincaré 56, 187 (1992) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Combescure, M., Robert, D.: Ann. Inst. H. Poincaré 61, 443 (1994) zbMATHMathSciNetGoogle Scholar
  7. 7.
    De Bièvre, S., Renaud, J.: J. Phys. A: Math. Gen. 29, L585 (1996) CrossRefGoogle Scholar
  8. 8.
    Duclos, P., Lev, O., Šťovíček, P., Vittot, M.: Rev. Math. Phys. 14, 531 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Feingold, M., Peres, A.: Phys. Rev. A 34, 591 (1986) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Gutzwiller, M.: J. Math. Phys. 12, 343 (1971) CrossRefGoogle Scholar
  11. 11.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, New York (1966) Google Scholar
  12. 12.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon, London (1958) zbMATHGoogle Scholar
  13. 13.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic, New York (1974) Google Scholar
  14. 14.
    Paul, T., Uribe, A.: Ann. Inst. H. Poincaré 59, 357 (1993) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Ripamonti, N.: J. Phys. A: Math. Gen. 29, 5137 (1996) zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Ripamonti, N.: Math. Phys. Electron. J. 4, 2 (1998) MathSciNetGoogle Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Academic, New York (1978) zbMATHGoogle Scholar
  18. 18.
    Robert, D.: Helv. Phys. Acta 71, 44 (1998) MathSciNetGoogle Scholar
  19. 19.
    Zelditch, S.: J. Funct. Anal. 94, 415 (1990) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Nuclear ScienceCzech Technical UniversityPragueCzech Republic

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