Asymptotic Lattices, Good Labellings, and the Rotation Number for Quantum Integrable Systems
Résumé
This article introduces the notion of good labellings for asymptotic
lattices in order to study joint spectra of quantum integrable
systems from the point of view of inverse spectral theory. As an
application, we consider a new spectral quantity for a quantum
integrable system, the quantum rotation number. In the case of two
degrees of freedom, we obtain a constructive algorithm for the
detection of appropriate labellings for joint eigenvalues, which we
use to prove that, in the semiclassical limit, the quantum rotation
number can be calculated on a joint spectrum in a robust way, and
converges to the well-known classical rotation number. The general
results are applied to the semitoric case where formulas become
particularly natural.
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