- #1

Squark

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