arXiv subjects

O. Bourget

Publications and source records attributed to O. Bourget.

4 recordsLinked to original sources

arXiv2015

Spectral stability for compact perturbations of Toeplitz matrices

Let $f$ be a regular real-valued non-constant symbol defined on the one dimensional torus ${\mathbb T}$. Denote respectively by $κ$ and $T$, its set of critical points and the associated Toeplitz matrix on $l^2({\mathbb N})$. If $V$ is a suitable compact perturbation, we prove that the operator $T+V$ has no singular continuous spectrum and only finite point spectrum away from the set of thresholds $f(κ)$. We also obtain some propagation estimates and apply these results to concrete examples.

math.FA↗

arXiv2015

Commutation Relations for Unitary Operators II

Let $f$ be a regular non-constant symbol defined on the $d$-dimensional torus ${\mathbb T}^d$ with values on the unit circle. Denote respectively by $κ$ and $L$, its set of critical points and the associated Laurent operator on $l^2({\mathbb Z}^d)$. Let $U$ be a suitable unitary local perturbation of $L$. We show that the operator $U$ has finite point spectrum and no singular continuous component away from the set $f(κ)$. We apply these results and provide a new approach to analyze the spectral properties of GGT matrices with asymptotically constant Verblunsky coefficients. The proofs are based on positive commutator techniques. We also obtain some propagation estimates.

math.FA↗

arXiv2013

Commutation Relations for Unitary Operators III

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary self-adjoint operator $A$ defined on ${\cal H}$ allows to prove that the spectrum of $U$ has no singular continuous spectrum and a finite point spectrum, at least locally. We prove that under stronger regularity hypotheses, the local regularity properties of the spectral measure of $U$ are improved, leading to a better control of the decay of the correlation functions. As shown in the applications, these results may be applied to the study of periodic time-dependent quantum systems, classical dynamical systems and spectral problems related to the theory of orthogonal polynomials on the unit circle.

math.FA↗

arXiv2013

Commutation Relations for Unitary Operators

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary self-adjoint operator $A$ defined on ${\cal H}$ allows to prove that the spectrum of $U$ has no singular continuous spectrum and a finite point spectrum, at least locally. 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.

math.FA↗