arXiv subjects
Publications and source records attributed to S. L. Carvalho.
We show that the power-law decay exponents in von Neumann's Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value~$1$. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann's Ergodic Theorem depend on sequences of time going to infinity.
In the present work we consider off-diagonal Jacobi matrices with uncertainty in the position of sparse perturbations. We prove (Theorem 3.2) that the sequence of Prüfer angles (θ_{k}^ω)_{k\geq 1} is u.d mod πfor all ϕ\in [0,π] with exception of the set of rational numbers and for almost every ωwith respect to the product ν=\prod_{j\geq 1}ν_{j} of uniform measures on {-j,...,j}. Together with an improved criterion for pure point spectrum (Lemma 4.1), this provides a simple and natural alternative proof of a result of Zlatos (J. Funct. Anal. \textbf{207}, 216-252 (2004)): the existence of pure point (p.p) spectrum and singular continuous (s.c.) spectra on sets complementary to one another with respect to the essential spectrum [-2,2], outside sets A_{sc} and A_{pp}, respectively, both of zero Lebesgue measure (Theorem 2.4). Our method allows for an explicit characterization of A_{pp}, which is seen to be also of dense p.p. type, and thus the spectrum is proved to be exclusively pure point on one subset of the essential spectrum.
We consider off-diagonal Jacobi matrices $J$ with (faster-than-exponential) sparse perturbations. We prove (Theorem \ref{onehalf}) that the Fourier transform $\hat{\left\| f\right\| ^{2}dρ}(t)$ of the spectral measure $ρ$ of $J$, whose sparse perturbations are at least separated by a distance $\exp \left(cj(\ln j)^{2}\right) /δ^{j}$, for some $c>1/2,$ $0<δ<1$ and for a dense subset of $C_{0}^{\infty}(-2,2)$-functions $f$, decays as $t^{-1/2}Ω(t)$, uniformly in the spectrum $[-2,2]$, $Ω(t)$ increasing less rapidly than any positive power of $t$, improving earlier results obtained by Simon (Commun. Math. Phys. \textbf{179}, 713-722 (1996)) and by Krutikov-Remling (Commun. Math. Phys. \textbf{223}, 509-532 (2001)) for Schrödinger operators with sparse potential that increases as fast as exponential-of-exponential. Applications to the spectrum of the Kronecker sum of two (or more) copies of the model are given.
We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i. e. nonrandom, block-Jacobi matrices may be determined exactly, improving a result of Zlatos (J. Funct. Anal. 207, 216-252 (2004)).