Quantum singular complete integrability
speaker: Thierry Paul (Centre de mathématiques Laurent Schwartz Ecole polytechnique )abstract: We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schrödinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant and therefore the result is valid at the classical level. The unperturbed frequencies satisfy a small divisors condition (Bruno type condition, including the Diophantine case) and we explicitly estimate how this condition can be released when the family tends to the unperturbed one.
timetable:
Tue 6 May, 15:30 - 16:30, Aula Dini
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