A Mathematical Justification for the Herman-Kluk Propagator



A class of Fourier Integral Operators which converge to the unitary group of the Schrödinger equation in the semiclassical limit ε → 0 in the uniform operator norm is constructed. The convergence allows for an error bound of order O(ε), which can be improved to arbitrary order in ε upon the introduction of corrections in the symbol. On the Ehrenfest-timescale, the result holds with a slightly weaker error bound. In the chemical literature the approximation is known as the Herman-Kluk propagator.


  1. BaGrPa99.
    Bambusi D., Graffi S., Paul T.: Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time. Asympt. Anal. 21, 149–160 (1999)MATHMathSciNetGoogle Scholar
  2. BiRo03.
    de Bièvre S., Robert D.: Semi-classical Propagation on |log ħ|-timescales. IMRN 12, 667–696 (2003)CrossRefGoogle Scholar
  3. BiRo01.
    Bily, J.-M., Robert, D.: The semi-classical Van-Vleck formula. Application to the Aharonov-Bohm effect, Graffi, S. ed., et al., In: Long time behaviour of classical and quantum systems, Proceedings of the Bologna APTEX international conference, Bologna, Italy, September 13–17, 1999, Ser. Concr. Appl. Math., 1, Singapore: World Scientific, 2001, pp.89–106Google Scholar
  4. Bo03.
    Bony, J-M.: Evolution equations and microlocal analysis. In: Hyperbolic problems and related topics, Grad. Ser. Anal., Somerville, MA: International Press, 2003, pp.17–40Google Scholar
  5. BoRo02.
    Bouzouina A., Robert D.: Uniform semiclassical estimates for the propagation of quantum observables. Duke Math. J. 111(2), 223–252 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. Bu02.
    Butler J.: Global h Fourier integral operators with complex-valued phase functions. Bull. London Math. Soc. 34(4), 479–489 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. Co68.
    Cole J.D.: Perturbation Methods in Applied Mathematics. Blaisdell Publishing Co., Waltham, MA (1968)MATHGoogle Scholar
  8. CoRo97.
    Combescure M., Robert D.: Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asympt. Anal. 14, 377–404 (1997)MATHMathSciNetGoogle Scholar
  9. Fo89.
    Folland G.B.: Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122. Princeton University Press, Princeton, NJ (1989)Google Scholar
  10. Fu79.
    Fujiwara D.: A construction of the fundamental solution for the Schrödinger equation. J. d’Analyses Math. 35, 41–96 (1979)MATHCrossRefMathSciNetGoogle Scholar
  11. Ha85.
    Hagedorn G.: Semiclassical quantum mechanics I: The ħ→ 0 limit for coherent States. Commun. Math. Phys. 71, 77–93 (1980)CrossRefADSMathSciNetGoogle Scholar
  12. Ha98.
    Hagedorn G.: Raising and Lowering Operators for Semiclassical Wave Packets. Ann. Phys. 269(1), 77–104 (1998)MATHCrossRefADSMathSciNetGoogle Scholar
  13. HaJo00.
    Hagedorn G., Joye A.: Exponentially accurate semi-classical dynamics: propagation, localization, Ehrenfest Times, Scattering and More General States. Ann. Henri Poincaré 1(5), 837–883 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. HaRoGr04.
    Harabati C., Rost J.M., Grossman F.: Long-time and unitarity properties of semiclassical initial value representations. J. Chem. Phys. 120(1), 16–30 (2004)CrossRefADSGoogle Scholar
  15. He75.
    Heller E.J.: Time-dependent approach to semiclassical dynamics. J. Chem. Phys. 62(4), 1544–1555 (1975)CrossRefADSGoogle Scholar
  16. He81.
    Heller E.J.: Frozen Gaussians: a very simple semiclassical approximation. J. Chem. Phys. 75(6), 2923–2931 (1981)CrossRefMathSciNetADSGoogle Scholar
  17. HeKl84.
    Herman M.F., Kluk E.: A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations. Chem. Phys. 91(1), 27–34 (1984)CrossRefGoogle Scholar
  18. Hö83.
    Hörmander L.: The Analysis of Linear Partial Differential Operators I. Springer-Verlag, New York (1983)MATHGoogle Scholar
  19. JoOkRe01.
    Johnson C.R., Okubo Y., Reams R.: Uniqueness of Matrix Square Roots and an Application. Lin. Alg. Appl. 323, 51–60 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. Ka94.
    Kay K.: Integral expressions for the semi-classical time-dependent propagator. J. Chem. Phys. 100(6), 4377–4392 (1994)CrossRefADSMathSciNetGoogle Scholar
  21. Ka06.
    Kay K.: The Herman-Kluk approximation: Derivation and semiclassical corrections. Chem. Phys. 322, 3–12 (2006)CrossRefADSGoogle Scholar
  22. Ka66.
    Kato T.: Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966)MATHGoogle Scholar
  23. Ki82.
    Kitada H.: A calculus of Fourier integral operators and the global fundamental solution for a Schrödinger equation. Osaka J. Math. 19, 863–900 (1982)MATHMathSciNetGoogle Scholar
  24. KiKu81.
    Kitada H., Kumano-Go H.: A family of Fourier Integral Operators and the fundamental solution for a Schrödinger equation. Osaka J. Math. 18, 291–360 (1981)MATHMathSciNetGoogle Scholar
  25. LaSi00.
    Laptev A., Sigal I.M.: Global Fourier Integral Operators and semiclassical asymptotics. Review of Math. Phys. 12(5), 749–766 (2000)MATHCrossRefADSMathSciNetGoogle Scholar
  26. Ma02.
    Martinez A.: An Introduction to Semiclassical and Microlocal Analysis, Universitext. Springer-Verlag, New York (2002)Google Scholar
  27. RoSw07.
    Rousse, V., Swart, T.: Global L 2-Boundedness Theorems for Semiclassical Fourier Integral Operators with Complex Phase. Submitted, preprint is available online at http://arxiv.org/abs/0710.4200, 2007
  28. Ta04.
    Tataru, D.: Phase space transforms and microlocal analysis. In: Phase space analysis of partial differential equations. Vol. II, Pisa: Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., 2004, pp. 505–524Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Freie Universität Berlin, Institut für MathematikBerlinGermany

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