Olivier Bourget

Spectral Analysis of Unitary Band Matrices

Abstract: This paper is devoted to the spectral properties of a class of unitary operators with a matrix representation displaying a band structure. Such band matrices appear as monodromy operators in the study of certain quantum dynamical systems. These doubly infinite matrices essentially depend on an infinite sequence of phases which govern their spectral properties. We prove the spectrum is purely singular for random phases and purely absolutely continuous in case they provide the doubly infinite matrix with a periodic structure in the diagonal direction. We also study some properties of the singular spectrum of such matrices considered as infinite in one direction only.

Localization Properties of the Chalker–Coddington Model

The Chalker–Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove first that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly, that this implies spectral localization. Thirdly, we prove a Thouless formula and compute the mean Lyapunov exponent, which is independent of M.

Spectral Stability of Unitary Network Models

We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one dimensional quantum walks are universal, as are CMV matrices. We prove spectral stability and propagation properties for general asymptotically uniform models by means of unitary Mourre theory.

Dynamical Localization of the Chalker-Coddington Model far from Transition

We study a quantum network percolation model which is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We show dynamical localization for parameters corresponding to edges of Landau bands, away from the expected transition point.

Singular continuous Floquet operator for systems with increasing gaps

Abstract Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e−iH0Te−iκT|φ〉〈φ|, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H0 is pure point, simple, bounded from below and the gaps between the eigenvalues (λn) grow like λn+1−λn∼Cnd with d⩾2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H0, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.

Commutation relations for unitary operators II

Let

Lower bounds for sojourn time in a simple shape resonance model

f

Energy-Time Uncertainty Principle and Lower Bounds on Sojourn Time

be a regular non-constant symbol defined on the

Resonances under rank-one perturbations

d

Stability of the Electron Cyclotron Resonance

-dimensional torus