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Mathematical Basis of Bohr-Sommerfeld

  1. Nov 4, 2006 #1
    Hello everyone!

    The Bohr-Sommerfeld rule allows approximating the spectra of integrable
    systems in a quite accurate way, at least for high quantum numbers.
    The most general formulation of the rule is as follows (I converted a
    less
    high-brow formulation into this form, I hope I got it right):

    We know the phase-space (X, omega) of an integrable system is foliated
    by invariant Lagrangian tori (btw, does it mean the phase-space is a
    locally trivial fibration with toric fiber? Or are there exceptional
    fibers?)
    Lets choose a U(1) bundle L and connection A over the phase space,
    such that omega is the curvature of A (together with the Lagrangian
    foliation we have for free, these comprise precisely the data for
    geometric quantization!) Then, the restriction of A on any of the the
    tori
    is flat (since the tori are Lagrangian). However, some tori are
    special:
    the restriction on them is not only flat but trivial (all of the
    monodromies
    are trivial). These are the tori "selected" by the Bohr-Sommerfeld
    rule,
    and the values of the Hamiltonian (and the other integrals of motion)
    on
    them form the predicted quantum spectrum.

    The question is, has anyone shown the approximation to be "good", in
    some sense, in a mathematically rigorous way? At least for the high
    quantum number asymptotics?

    Best regards,
    Squark
     
  2. jcsd
  3. Nov 4, 2006 #2
    Squark wrote:


    > We know the phase-space (X, omega) of an integrable system is foliated
    > by invariant Lagrangian tori (btw, does it mean the phase-space is a
    > locally trivial fibration with toric fiber? Or are there exceptional
    > fibers?)


    There can be singular fibers, in fact this is the case already in many
    simple examples of integrable systems (see below).

    > The question is, has anyone shown the approximation to be "good", in
    > some sense, in a mathematically rigorous way? At least for the high
    > quantum number asymptotics?


    Yes this has been shown (e.g. section 5 of the second reference below).

    The study of Bohr-Sommerfeld rules in the case of integrable systems
    with singularities has received a lot of attention recently from both
    mathematicians and physicists, in relation to the idea of "quantum
    monodromy" (basically the lattice formed by the quantum states is
    usually not a simple periodic one and the quantum numbers are not
    globally valid).

    Here are three recent references which provide some background:

    1. Singular Bohr-Sommerfeld rules for 2D integrable systems
    Yves Colin de Verdière and San Vu Ngoc
    Annales Scientifiques de l'École Normale Supérieure
    Volume 36, Issue 1 , March 2003, Pages 1-55.
    doi:10.1016/S0012-9593(03)00002-8
    preprint: http://www-fourier.ujf-grenoble.fr/~svungoc/articles/bs2d.pdf

    2. Symplectic Techniques for Semiclassical Integrable Systems
    San Vu Ngoc
    2004
    preprint:
    http://www-fourier.ujf-grenoble.fr/~svungoc/articles/stsis.pdf

    3. Hamiltonian monodromy as lattice defect
    B. I. Zhilinskií
    in: Topology in Condensed Matter,
    (Springer Series in Solid-State Sciences, Vol. 150), 2006, pp. 165-186.
    preprint: http://pca3.univ-littoral.fr/~zhilin/prepub/MLD.ps


    Experimentalists are also starting to study the issue, see

    4. Experimental Confirmation of Quantum Monodromy: The Millimeter Wave
    Spectrum of Cyanogen Isothiocyanate NCNCS
    B. P. Winnewisser at al.
    Phys. Rev. Lett. 95, 243002 (2005)
    http://link.aps.org/abstract/PRL/v95/e243002


    Regards,
    ---
    PP
     
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