Commutator methods for unitary operators

Abstract

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 niteness 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 rst time that commutator methods are applied in the context of rotation algebras, for the study of their generators.

Quantum walks with an anisotropic coin I: spectral theory

“…In the one-Hilbert space setting, the unitary operator U is usually a multiplicative perturbation of the unitary operator U 0 . In this case, if U − U 0 is compact, the stability of the function ̺ A0 U0 under compact perturbations allows one to infer information on U from similar information on U 0 (see [12,Cor. 2.10]).…”

“…The claim follows by adapting the proof of [12, Prop. 2.9] to locally U-smooth operators T with values in the auxiliary Hilbert space G, taking into account the results of Section 3.2.The last theorem of this section corresponds to[12, Thm. 2.7]: Spectrum of U).…”

The absolute continuous spectrum of skew products of compact Lie groups

“…(the operator A N is self-adjoint on D(A N ) = D(A) because (U π,j ) n ∈ C 1 (A) for each n ∈ Z, see [10,Sec. 4]).…”

Quantum time delay for unitary operators: General theory

“…In Section 3, we use commutator methods for unitary operators [9] to construct a conjugate operator for U 0 and to prove a Mourre estimate for U 0 on the set σ(U 0 ) \ κ(U 0 ) (Lemmas 3.1 and 3.3). As a consequence, we obtain in Theorem 3.4 a class locally U 0 -smooth operators on S 1 \ κ(U 0 ) and we show that the operator U 0 has purely absolutely continuous in σ(U 0 ) \ κ(U 0 ).…”