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César R. de Oliveira

Publications and source records attributed to César R. de Oliveira.

12 recordsLinked to original sources
arXiv2026

Quantitative spectral stability and sharp Fourier estimates for periodic CMV matrices with decaying perturbations

We study half-line CMV matrices whose Verblunsky coefficients are decaying perturbations of periodic sequences. We quantify the stability of fine properties of the absolutely continuous spectral measure under such perturbations. More precisely, we establish Hölder continuity of the perturbed spectral measure and derive sharp time-averaged Fourier estimates governing the associated quantum dynamics. These results remain valid up to the edges of the absolutely continuous spectrum. Since the spectral measure of a half-line CMV matrix is precisely the orthogonality measure for the corresponding orthogonal polynomials on the unit circle (OPUC), our results also yield new statements for OPUC whose Verblunsky coefficients are decaying perturbations of periodic sequences.

math.SP↗
arXiv2022

On the Birman-Krein Theorem

It is shown that if $X$ is a unitary operator so that a singular subspace of~$U$ is unitarily equivalent to a singular subspace of~$UX$ (or $XU$), for each unitary operator~$U$, then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman-Krein Theorem that includes the preservation of a singular spectral subspace in this context.

math-ph↗
arXiv2022

Slow dynamics for self-adjoint semigroups and unitary evolution groups

We obtain slow dynamics for self-adjoint semigroups and unitary evolution groups. For semigroups, the slow dynamics is for orbits, and for the average return probability in the case of unitary evolution groups. We present an application to the quantum dynamics of purely absolutely continuous systems

math.SP↗
arXiv2021

Some generic fractal properties of bounded self-adjoint operators

We study generic fractal properties of bounded self-adjoint operators through lower and upper generalized fractal dimensions of their spectral measures. Two groups of results are presented. Firstly, it is shown that the set of vectors whose associated spectral measures have lower (upper) generalized fractal dimension equal to zero (one) for every $q>1$ ($0<q<1$) is either empty or generic. The second one gives sufficient conditions, for separable regular spaces of operators, for the presence of generic extreme dimensional values; in this context, we have a new proof of the celebrated Wonderland Theorem.

math.SP↗
arXiv2021

Generic zero-Hausdorff and one-packing spectral measures

For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneously zero upper-Hausdorff and one lower-packing dimensions contains a dense $G_δ$ subset. Applications include sets of limit-periodic operators.

math.FA↗
arXiv2020

Hydrogen atom bound states whose spectral measures have positive upper fractal dimensions

It is shown that, Baire generically, the bound states of the Hamiltonian of the Hydrogen atom have spectral measures with exact $0$-lower and $1/3$-upper generalized fractal dimensions; the relation to (a weak form of) dynamical delocalization along orthonormal bases is also discussed. Such result is a consequence of the distribution of the Hamiltonian eigenvalues.

math-ph↗
arXiv2019

Category theorems for Schrödinger semigroups

Stimulated by the category theorems of Eisner and Serény in the setting of unitary and isometric $C_0$-semigroups on separable Hilbert spaces, we prove category theorems for Schrödinger semigroups. Specifically, we show that, to a given class of Schrödinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the corresponding Schrödinger operators.

math.SP↗
arXiv2019

Quantum quasiballistic dynamics and thick point spectrum

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schrödinger operators.

math-ph↗
arXiv2016

A characterization of singular packing subspaces with an application to limit-periodic operators

A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a $G_δ$ (in suitable metric spaces). As an application, it is proven that, generically (in space of continuous sampling functions), spectral measures of the limit-periodic Schrödinger operators have upper packing dimensions equal to one. Consequently, in a generic set, these operators are quasiballistic.

math-ph↗