Quantum singular complete integrability

Thierry Paul (1) , Laurent Stolovitch (2)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 LJAD - Laboratoire Jean Alexandre Dieudonné

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 near $\hbar= 0$. 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.

Mots clés: équations aux dérivées partielles; systèmes dynamiques; mécanique quantique; série de Birkhoff; théorie des perturbations

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Thierry Paul, Laurent Stolovitch. Quantum singular complete integrability. Journal of Functional Analysis, 2016, 271, pp.1377-1443. ⟨hal-00945409v2⟩