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Silas L. Carvalho

Publications and source records attributed to Silas L. Carvalho.

At least 19 recordsLinked to original sources
arXiv2026

Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under exponentially decaying perturbations

We establish $\frac{1}{2}$-Hölder continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schrödinger operators under suitable boundary conditions and sufficiently fast polynomially decaying small potentials. These are the first known examples, apart from the free case, of Schrödinger operators with Lipschitz continuous spectral measures up to the spectral edge, a property obtained here as a consequence of the Dirichlet boundary condition. Notably, we show that the asymptotic behavior of the time-averaged quantum return probability, either $\log (t) / t$ or $1 / t$, as in the case of the free Laplacian, remains unchanged in this setting. Furthermore, we prove the persistence of the purely absolutely continuous spectrum and the $\frac{1}{2}$-Hölder continuity of the spectral measures for (Diophantine) quasi-periodic operators under exponentially decaying small perturbations. These results are optimal and hold for all energies, up to the border of the absolutely continuous spectrum.

math.SP↗
arXiv2026

The Oracle Theorem for Matrix-Valued Jacobi Operators

This paper develops the matrix-valued analogue of the reflectionless and oracle framework for Jacobi operators. Starting from matrix-valued Weyl--Titchmarsh $m$-functions on the Siegel upper half-space, we study the distance-decreasing action of transfer matrices, matrix-valued harmonic measures and value distribution convergence. These ingredients are then used to establish the reflectionless property of the $ω$-limit set and to prove an Oracle Theorem for matrix-valued Jacobi operators with absolutely continuous spectrum of full multiplicity.

math-ph↗
arXiv2025

Exponential dichotomy for dynamically defined matrix-valued Jacobi operators

We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $(\mathbb{C}^{l})^{\mathbb{Z}}$, given for each $ω\in Ω$ by the law $[H_ω \textbf{u}]_{n} := D(T^{n - 1}ω) \textbf{u}_{n - 1} + D(T^{n}ω) \textbf{u}_{n + 1} + V(T^{n}ω) \textbf{u}_{n}$, where $Ω$ is a compact metric space, $T: Ω\rightarrow Ω$ is a minimal homeomorphism and $D, V: Ω\rightarrow M(l, \mathbb{R})$ are continuous maps with $D(ω)$ invertible for each $ω\inΩ$. Namely, we show that for each $ω\inΩ$, \[ρ(H_ω)=\{z \in \mathbb{C}\mid (T, A_z)\;\mathrm{is\; uniformly\; hyperbolic}\}, \] where $ρ(H_ω)$ is the resolvent set of $H_ω$ and $(T, A_z)$ is the $SL(2l,\mathbb{C})$-cocycle induced by the eigenvalue equation $[H_ωu]_n=zu_n$ at $z\in\mathbb{C}$.

math.DS↗
arXiv2024

Generic dimensional and dynamical properties of invariant measures of full-shift systems over countable alphabets

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a countable set. More specifically, we show that the set of invariant measures with infinite packing dimension equal to infinity is a dense $G_δ$ subset of $\mathcal{M}(T)$, the space of $T$-invariant measures endowed with the weak topology, where the alphabet $M$ is a countable Polish metric space. We also show that the set of invariant measures with upper $q$-generalized fractal dimension (with $q>1$) equal to infinity is a dense $G_δ$ subset of $\mathcal{M}(T)$, where the alphabet $M$ is a countable compact metric space. This improves the results obtained by Carvalho and Condori in \cite{AS} and \cite{AS2}, respectively. Furthermore, we discuss the dynamical consequences of such results, regarding the upper recurrence rates and upper quantitative waiting time indicator for typical orbits, and how the fractal dimensions of invariant measures and such dynamical quantities behave under an $α$-Hölder conjugation.

math.DS↗
arXiv2023

Refined decay rates of $C_0$-semigroups on Banach spaces

We study rates of decay for $C_0$-semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with $\vert s\vert^β\log(\vert s\vert)^b$, $β, b \geq 0$, as $\vert s\vert\rightarrow\infty$, and with $\vert s\vert^{-α}\log(1/\vert s\vert)^a$, $α, a \geq 0$, as $\vert s\vert \rightarrow 0$. Our results do not suppose that the semigroup is bounded. In particular, for $a=b=0$, our results improve the rates involving Fourier types obtained by Rozendaal and Veraar (J. Funct. Anal. 275(10): 2845-2894, 2018).

math.FA↗
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↗
arXiv2022

Criteria for the Absolutely Continuous Spectral Components of matrix-valued Jacobi operators

We extend in this work the Jitomirskaya-Last inequality and Last-Simoncriterion for the absolutely continuous spectral component of a half-line Schrödinger operator to the special class of matrix-valued Jacobi operators $H:l^2(\mathbb{Z},\mathbb{C})\rightarrow l^2(\mathbb{Z},\mathbb{C})$ given by the law $[H \textbf{u}]_{n} := D_{n - 1} \textbf{u}_{n - 1} + D_{n} \textbf{u}_{n + 1} + V_{n} \textbf{u}_{n}$, where $(D_n)_n$ and $(V_n)_n$ are bilateral sequences of $l\times l$ self-adjoint matrices such that $0<\inf_{n\in\mathbb{Z}}s_l[D_n]\le\sup_{n\in\mathbb{Z}}s_1[D_n]<\infty$ (here, $s_k[A]$ stands for the $k$-th singular value of $A$). Moreover, we also show that the absolutely continuous components of even multiplicity of minimal dynamically defined matrix-valued Jacobi operators are constant, extending another result from Last-Simon originally proven for scalar Schrödinger operators.

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

Kotani Theory for ergodic matrix-like Jacobi operators

We extend the so-called Kotani Theory for a particular class of ergodic matrix-like Jacobi operators defined in $l^{2}(\mathbb{Z}; \mathbb{C}^{l})$ by the law $[H_ω \textbf{u}]_{n} := D^{*}(T^{n - 1}ω) \textbf{u}_{n - 1} + D(T^{n}ω) \textbf{u}_{n + 1} + V(T^{n}ω) \textbf{u}_{n}$, where $T: Ω\rightarrow Ω$ is an ergodic automorphism in the measure space $(Ω, ν)$, the map $D: Ω\rightarrow GL(l, \mathbb{R})$ is bounded, and for each $ω\inΩ$, $D(ω)$ is symmetric. Namely, it is shown that for each $r\in\{1,\ldots,l\}$, the essential closure of $\mathcal{Z}_{r} := \{x \in \mathbb{R}\mid$ exactly $2r$ Lyapunov exponents of $A_z$ are zero$\}$ coincides with $σ_{ac,2r}(H_ω)$, the absolutely continuous spectrum of multiplicity $2r$, where $A_z$ is a Schrödinger-like cocycle induced by $H_ω$. Moreover, if $k\in\{1,\ldots,2l\}$ is odd, then $σ_{ac,k}(H_ω)=\emptyset$ for $ν$-a.e. $ω\inΩ$. We also provide a Thouless Formula for such class of operators.

math-ph↗
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

On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces

In this work, we show that if $f$ is a uniformly continuous map defined over a Polish metric space, then the set of $f$-invariant measures with zero metric entropy is a $G_δ$ set (in the weak topology). In particular, this set is generic if the set of $f$-periodic measures is dense in the set of $f$-invariant measures. This settles a conjecture posed by Sigmund (Sigmund, K. On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285-299) which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if $X$ is compact and if $f$ is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for $q\in(0,1)$ is equal to zero. Moreover, we show that if $X$ is a compact metric space and if $f$ is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equal to zero is residual. Finally, we present an alternative proof of the fact that the set of expansive measures is a $G_{δσ}$ set in the set of probability measures $\M(X)$, if $X$ is a Polish metric space and if $f$ is uniformly continuous; this result was originally proved by Lee, Morales and Shin (Lee, K., Morales, C. A., and Shin, B. On the set of expansive measures. Commun. Contemp. Math. 20, 7 (2018), 1750086, 10) for compact metric spaces.

math.DS↗
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

A note on the relation between the metric entropy and the generalized fractal dimensions of invariant measures

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower $q$-generalized fractal dimensions $D^{\pm}_μ(q)$, $q\in\mathbb{R}$, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem~\cite{Young} for the generalized fractal dimensions of the Bowen-Margulis measure associated with a $C^{1+α}$-Axiom A system over a two-dimensional compact Riemannian manifold $M$. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures such that $D_μ^+(q)=0$ ($q\ge 1$), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\in [0,1)$, $D^{+}_μ(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in~\cite{Sigmund1974} for Lipschitz transformations which satisfy the specification property.

math.DS↗
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

Site recurrence of open and unitary quantum walks on the line

We study the problem of site recurrence of discrete time nearest neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks our discussion concerns two notions of recurrence, one given by a monitoring procedure, another in terms of Pólya numbers, and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks are related by an additive interference term in a simple way. Based on a previous result of positive recurrence we describe an open quantum version of Kac's lemma for the expected return time to a site.

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↗