Commutation relations for unitary operators I

Abstract

Let U be a unitary operator defined on some infinite-dimensional complex Hilbert space H. Under some suitable regularity assumptions, it is known that a positive commutation relation between U and an auxiliary self-adjoint operator A defined on H allows to prove that the spectrum of U has locally no singular continuous spectrum and a finite point spectrum. We show that these conclusions still hold under weak regularity hypotheses and without any gap condition. As an application, we study the spectral properties of the Floquet operator associated to some perturbations of the quantum harmonic oscillator under resonant AC-Stark potential.