Advertisement

Semiclassical Normal Forms

  • D. Bambusi
Conference paper
Part of the Trends in Mathematics book series (TM)

Summary

Given a classical Hamiltonian function having an absolute minimum, we consider the problem of describing in the semiclassical limit the lowest part of the spectrum of the corresponding quantum operator. To this end we present an extension of the classical Birkhoff normal form to the semiclassical context and we use it to deduce spectral information on the quantum Hamiltonian. The properties of the spectrum turn out to be strongly dependent on the resonance relations fulfilled by the frequencies of small oscillations of the classical system. Here we concentrate on two opposite cases, namely the completely nonresonant and the completely resonant one and describe the spectrum of the Hamiltonian in these cases.

Keywords

Normal Form Toeplitz Operator Unitary Transformation Hamiltonian Operator Principal Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bam95]
    D. Bambusi, Uniform Nekhoroshev estimates on quantum normal forms Nonlinearity. 8 (1995), 93–105.MathSciNetMATHCrossRefGoogle Scholar
  2. [BCTO3]
    D. Bambusi, L. Charles and S. Tagliaferro, Resonant normal forms and splitting of degenerate eigenvalues, preprint, 2003.Google Scholar
  3. [BGP99]
    D. Bambusi, S. Graffi, T. Paul, Normal forms and quantization formulae Commun. Math. Phys. 207 (1999), 173–195.MathSciNetMATHCrossRefGoogle Scholar
  4. [BV90]
    J. Bellissard, M. Vittot, Heisenberg picture and noncommutative geometry of classical limit in quantum mechanics, Ann. Inst. H. Poincaré. 52 (1990), 175–235.MathSciNetMATHGoogle Scholar
  5. [BS91]
    F.A. Berezin and M.S. Shubin The Schrödinger Equation Kluwer, 1991.CrossRefGoogle Scholar
  6. [Cha02]
    L. Charles, Toeplitz operators and symplectic reduction, preprint, 2002.Google Scholar
  7. [GP87]
    S. Graffi, T. Paul, The Schrödinger equation and canonical perturbation theory Commun. Math. Phys. 108 (1987), 25–40.MathSciNetMATHCrossRefGoogle Scholar
  8. [Pop00a]
    G. Popov, Invariant tori effective stability and quasimodes with expo-nentially small error terms I, Ann. Henri Poincaré 1 (2000), 223–248.Google Scholar
  9. [Pop00b]
    G. Popov, Invariant tori effective stability and quasimodes with expo-nentially small error terms II Ann. Henri Poincaré 1 (2000), 249–279.MATHCrossRefGoogle Scholar
  10. [Ro87]
    D. Robert Autour de l’approximation semiclassique Birkhäuser, Basel,1987.Google Scholar
  11. [Sjo92]
    J. SJÖstrand, Semi-excited levels in non-degenerate potential wells Asymptotic Analysis 6 (1992), 29–43.MathSciNetMATHGoogle Scholar
  12. [Vun98]
    S. Vú NGOC, Sur le spectre des systemes complètement intégrables semi-classiques avec singularités, thesis, Institut Fourier, 1998.Google Scholar
  13. [VSZBO1]
    CH. van Hecke, D. Sadovskii, B. I. Zhilinskiì, V. Boudon, Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule P4 Europ. Phys. J. D 17 (2001), 13–35.Google Scholar
  14. [Zhi89]
    B. Zhilinskiì Chem. Phys. 137 (1989), 1–13.Google Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • D. Bambusi
    • 1
  1. 1.Dipartimento di MatematicaMilanoItaly

Personalised recommendations