The inverse spectral problem for quantum semitoric systems
Résumé
Given a quantum semitoric system composed of pseudodifferential
operators, Berezin-Toeplitz operators, or a combination of both, we
obtain explicit formulas for recovering, from the semiclassical
asymptotics of the joint spectrum, all symplectic invariants of the
underlying classical semitoric system.
Our formulas are based on the possibility to obtain good quantum
numbers for joint eigenvalues from the bare data of the joint
spectrum. In the spectral region corresponding to regular values of
the momentum map, the algorithms developed by Dauge, Hall and the
second author [27] produce such
labellings. In our proof, it was crucial to extend these algorithms
to the boundary of the spectrum, which led to the new notion of
asymptotic half-lattices, and to globalize the resulting labellings.
Using the construction given by Pelayo and the second author
in [79], our results prove that semitoric systems
are completely spectrally determined in an algorithmic way~: from
the joint spectrum of a quantum semitoric system one can construct a
representative of the isomorphism class of the underlying classical
semitoric system. In particular, this recovers the uniqueness result
obtained by Pelayo and the authors
in [62,61], and completes
it with the explicit computation of all invariants, including the
twisting index.
In the cases of the spin-oscillator and the coupled angular momenta,
we implement the algorithms and illustrate numerically the
computation of the invariants from the joint spectrum.
Mots clés
Semitoric integrable systems
Inverse spectral theory
Quantum mechanics
Semiclassical analysis
Pseudodifferential operators
Berezin-Toeplitz operators
Symplectic invariants
Focus-focus singularity
Joint spectrum
Asymptotic lattice
Good labelling
Quantum numbers
Lattice structure detection
Singular Lagrangian fibration
Origine | Fichiers produits par l'(les) auteur(s) |
---|