SEMINAIRE VENDREDI 30 JANVIER 2004
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Prof. Claudio Fernandez
Facultad de Matematicas,
Univ. Pontifica Catholica del Chile, Santiago, Chile

Title: Commutator estimates for unitary operators

Abstract: The scope of this talk is to generalize the Mourre' estimates (or
local commutators estimates) for self-adjoint operators. The
first of such results is the Putman' theorem which can be used to
prove absolute continuity of some self-adjoint operators, from a
positive commutator condition. We extend this result for a unitary
operator U and then we discuss the connection with smoothness
and with sojourn time. Also, we shall discuss how to obtain
results on the spectrum of U from estimates on the commutator of
U with adequate operators.

Our main application concerns the time dependant Schrodinger
equation
i\frac{d\varphi}{dt}= (H_0+V(t)\varphi,
where H_0 is a fixed self-adjoint operator. In the periodic
case V(t+T) = V(t), the asymptotic behavior of the solutions
can be described by the operator giving the evolution after one
period. This one is given by a unitary
operator U, the nature of whose spectrum is related to the
dynamical properties of these solutions.

Above is part of a joint work with prof. M.A.Astaburuaga,
O.Bourget and V.Cortes.