Communications in Mathematical Physics

, Volume 203, Issue 2, pp 465–479 | Cite as

Quantum Monodromy in Integrable Systems

  • San Vũ Ngoc


Let P 1(h),...,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 Verdière [6] and Charbonnel [3] 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 [9]. 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 [7] concerning the quantum spherical pendulum.


Manifold Integrable System Principal Symbol Integral Lattice Joint Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • San Vũ Ngoc
    • 1
  1. 1.Institut Fourier UMR5582, B.P. 74, 38402 Saint-Martin d'Hères, France.¶E-mail: San.Vu-Ngoc@ujf-grenoble.frFR

Personalised recommendations