The Born-Oppenheimer Expansion: Eigenvalues, Eigenfunctions and Low-Energy Scattering
- 179 Downloads
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.
KeywordsFormal Power Series Diatomic Molecule Pseudo Differential Operator Wave Operator Spectral Projection
Unable to display preview. Download preview PDF.
- 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
- 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
- B. Helffer and J. Sjostrand. Puits multiple en mecanique semi-classique 6. Ann. Inst. H. Poinc., 46: 353–372, 1987.Google Scholar
- W. Hunziker. Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poinc., 45: 339–358, 1986.Google Scholar
- H. Isozaki and H. Kitada. Modified wave operators with time dependent modifiers. J. Fac. Sci. Univ. Tokyo, 32: 77–104, 1985.Google Scholar
- M. Klein, A. Martinez, and X. P. Wang. On the born-oppenheimer approx-imation of wave operators for singular potentials, preprint, 1995.Google Scholar
- A. Martinez and B. Messirdi. Resonances of diatomic molecules in the born-oppenheimer approximation, preprint, 1992.Google Scholar
- P. Pettersson. WKB expansions for systems of schrodinger operators with crossing eigenvalues. Preprint, Lund University, 1993.Google Scholar
- D. Robert. Autour de 1’ approximation semiclassique. Birkhauser, Progress in Math., 68, 1987.Google Scholar
- B. Simon and M. Reed. Methods of modern mathematical physics 3. Scat-tering theory. Academic Press, 1979.Google Scholar
- B. R. Vainberg. Quasi-classical approximation in stationary scattering prob-lems. Fune. Anal. Appi, 12: 247–257, 1977.Google Scholar