The Born-Oppenheimer Expansion: Eigenvalues, Eigenfunctions and Low-Energy Scattering

  • Markus Klein
Conference paper
Part of the Lecture Notes in Chemistry book series (LNC, volume 71)


In these notes I want to review some work of A. Martinez, X.P. Wang, R. Seiler and myself on the rigorous justification of the Born-Oppenheimer approximation [10, 11, 12]. The results concern firstly complete asymptotic expansions of eigenvalues and eigenfunctions of WKB-type for the full quantum mechanical Hamiltonian of a polyatomic molecule. We denote the semiclassical expansion parameter by h; its square is essentially the quotient of electronic to nuclear mass. Already Born and Oppenheimer considered (formally) complete asymptotic expansions which, however, did not include the exponential weights of WKB-type expansions describing the exponential localisation of eigenfunctions near the bottom of the potential well formed by the first eigenvalue of the electronic Hamiltonian. In honour of their seminal paper we call our results BO-expansions, although some authors might use the word Born-Oppenheimer approximation in some much more restricted sense.


Formal Power Series Diatomic Molecule Pseudo Differential Operator Wave Operator Spectral Projection 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Chantelau. On the born-oppenheimer approximation for excited states of polyatomic schrodinger operators. Letters in Mathematical Physics, 25: 227–238, 1992.CrossRefGoogle Scholar
  2. [2]
    J. M. Combes, P. Duclos, and R. Seiler. The born-oppenheimer approxi-mation. Rig. atomic and molecular physics. Velo, Wightman (eds), pages 185–212, 1981.Google Scholar
  3. [3]
    C. Gerard and A. Martinez. Principe d’ absorption limite limite pour des operateurs de schrodinger a longue portee. C. R. Acad. Sci., 306: 121–123, 1988.Google Scholar
  4. [4]
    G. A. Hagedorn. High order corrections to the time-independent born-oppenheimer approximation 2. diatomic coulomb systems. Comm. Math. Phys., 116: 23–44, 1988.CrossRefGoogle Scholar
  5. [5]
    B. Helffer and J. Sjostrand. Multiple wells in the semiclassical limit 1. Comm. Part. Diff. Equ., 9: 337–408, 1984.CrossRefGoogle Scholar
  6. [6]
    B. Helffer and J. Sjostrand. Puits multiple en mecanique semi-classique 6. Ann. Inst. H. Poinc., 46: 353–372, 1987.Google Scholar
  7. [7]
    W. Hunziker. Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poinc., 45: 339–358, 1986.Google Scholar
  8. [8]
    H. Isozaki and H. Kitada. Modified wave operators with time dependent modifiers. J. Fac. Sci. Univ. Tokyo, 32: 77–104, 1985.Google Scholar
  9. [9]
    M. Klein. On the mathematical theory of predissociation. Ann. Phys., 178: 48–73, 1987.CrossRefGoogle Scholar
  10. [10]
    M. Klein, A. Martinez, R. Seiler, and X. P. Wang. On the born-oppenheimer approximation for polyatomic molecules. Comm. Math. Phys., 143: 607–639, 1992.CrossRefGoogle Scholar
  11. [11]
    M. Klein, A. Martinez, and X. P. Wang. Born-oppenheimer approximation of wave-operators. Comm. Math. Phys., 152: 73–95, 1993.CrossRefGoogle Scholar
  12. [12]
    M. Klein, A. Martinez, and X. P. Wang. On the born-oppenheimer approx-imation of wave operators for singular potentials, preprint, 1995.Google Scholar
  13. [13]
    A. Martinez. Resonances dans l’approximation de born-oppenheimer 1. J. Diff. Equ., 91: 204–234, 1991.CrossRefGoogle Scholar
  14. [14]
    A. Martinez and B. Messirdi. Resonances of diatomic molecules in the born-oppenheimer approximation, preprint, 1992.Google Scholar
  15. [15]
    P. Pettersson. WKB expansions for systems of schrodinger operators with crossing eigenvalues. Preprint, Lund University, 1993.Google Scholar
  16. [16]
    D. Robert. Autour de 1’ approximation semiclassique. Birkhauser, Progress in Math., 68, 1987.Google Scholar
  17. [17]
    D. Robert and H. Tamura. Asymptotic behaviour of scattering amplitudes in semi-classical and low energy limit. Ann. Inst. Fourier, 39: 155–192, 1989.CrossRefGoogle Scholar
  18. [18]
    B. Simon and M. Reed. Methods of modern mathematical physics 3. Scat-tering theory. Academic Press, 1979.Google Scholar
  19. [19]
    B. R. Vainberg. Quasi-classical approximation in stationary scattering prob-lems. Fune. Anal. Appi, 12: 247–257, 1977.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Markus Klein
    • 1
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

Personalised recommendations