Commutator methods for unitary operators

Claudio Fernandez (1) , Serge Richard (2) , Rafael Tiedra de Aldecoa (1)

1 Facultad de Matemáticas [Santiago de Chile]
2 PSPM - Probabilités, statistique, physique mathématique

We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generators.

Mots clés: Unitary operators; spectral analysis; Mourre theory; cocycles over rotations

Domaines: Physique mathématique [math-ph]

Claudio Fernandez, Serge Richard, Rafael Tiedra de Aldecoa. Commutator methods for unitary operators. J. Spectr. Theory, 2013, 3, pp.271 - 292. ⟨hal-00660531⟩