Vu Ngoc, S.
Quantum monodromy in integrable systems
(267K, postscript)
ABSTRACT. Let $P_1(h),\dots,P_n(h)$ be a set of commuting self-adjoint
$h$-pseudo-differential operators on an $n$-dimensional manifold. If
the joint principal symbol $p$ is proper, it is known from the work of
Colin de Verdi\`ere and Charbonnel that in a neighbourhood of any
regular value of $p$, the joint spectrum locally has the structure of
an affine integral lattice. This leads to the construction of a
natural invariant of the spectrum, called the quantum monodromy. We
present this construction here, and show that this invariant is given
by the classical monodromy of the underlying Liouville integrable
system, as introduced by Duistermaat. The most striking application of
this result is that all two degree of freedom quantum integrable
systems with a focus-focus singularity have the same non-trivial
quantum monodromy. For instance, this proves a conjecture of Cushman
and Duistermaat concerning the quantum spherical pendulum.