Commutator criteria for strong mixing

Abstract

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^N\}_{N\in\mathbb Z}$ and continuous flows $\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space $\mathcal H$. Our approach put into evidence a general definition for the topological degree of the curves $N\mapsto U^N$ and $t\mapsto{\rm e}^{-itH}$ in the unitary group of $\mathcal H$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on graphs.Comment: 15 pages, 1 figur

Decay estimates for unitary representations with applications to continuous- and discrete-time models

“…So, we infer from the assumption and Cauchy-Schwarz inequality that In the next proposition, we consider the particular case where [iH, A] = f (H) for some Borel function f : R → R. Our results in this case generalise the results of [36,].…”

“…In recent papers [5,26,29,[33][34][35][36][37][38], it has been shown that one can combine certain tools from dynamical systems (averaging along the dynamics, ergodic theorems) and quantum mechanics (commutator methods) to determine spectral properties of various classes of continuous-and discrete-time models. In particular, a new criterion for strong mixing has been put into evidence in [26,34].…”

“…The vector field f X 1 admits a complete flow F 1 := ( F 1,t ) t∈R uniquely ergodic with respect to the measure µ Ω := f −1 µΩ M f −1 dµΩ and with generator H := f H 1 essentially self-adjoint on C 1 (M ) ⊂ H := L 2 (M, f −1 µ Ω ). Furthermore, the following holds true [36,Ex. 4.8]: The operator…”

“…This example is motivated by [33,36], but see also [34,37]. Let Σ be a finite volume Riemann surface of genus ≥ 2 and let M := T 1 Σ be the unit tangent bundle of Σ.…”

Furstenberg Transformations on Cartesian Products of Infinite-Dimensional Tori

“…We recall in this section some facts borrowed from [7,14] on commutator methods for unitary operators and regularity classes associated with them.…”