KAM for the quantum harmonic oscillator

Benoît Grébert (1) , Laurent Thomann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.

Domaines: Equations aux dérivées partielles [math.AP]

Benoît Grébert, Laurent Thomann. KAM for the quantum harmonic oscillator. Communications in Mathematical Physics, 2011, 307 (2), pp.383-427. ⟨10.1007/s00220-011-1327-5⟩. ⟨hal-00464193⟩