Mathematical Basis of Bohr-Sommerfeld

[Squark]

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?Squark

 

[Pindare]

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?)[/color]

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?[/color]

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.psExperimentalists 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/e243002Regards,
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PP