Quasi-periodic Jacobi Operators Research Articles

Anderson localization for multi-frequency quasi-periodic Jacobi operators

<p style='text-indent:20px;'>In this paper, we are concerned with the quasi-periodic Jacobi operators with large potentials on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{T}^d $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ d \geq 1 $\end{document}</tex-math></inline-formula>) and establish the positivity and continuity of the Lyapunov exponent by combining the large deviation theorem with the avalanche principle. Moreover, we show that Anderson localization takes place for almost all Diophantine frequencies when the coupling is sufficiently large.</p>

Spectral dimension for β-almost periodic singular Jacobi operators and the extended Harper’s model

We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper’s model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents.

Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Russmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Russmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

Full measure reducibility and localization for quasiperiodic Jacobi operators: A topological criterion

We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasiperiodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region.

Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators

Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators

Spectral Approximation for Quasiperiodic Jacobi Operators

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into the associated quantum dynamics, that is, the one-parameter unitary group that solves the time-dependent Schrodinger equation. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary for detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-K Jacobi operator in O(K 2) operations, then use the algorithm to investigate the spectra of Schrodinger operators with Fibonacci, period doubling, and Thue–Morse potentials.

Dominated splittings and the spectrum of quasi-periodic Jacobi operators

We prove that the resolvent set of any, possibly singular, quasi- periodic Jacobi operator is characterized as the set of all energies whose associated Jacobi cocycles induce a dominated splitting. This extends a well-known result by Johnson for Schrödinger operators.

Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators

We consider the quasi-periodic Jacobi operator $H_{x,\omega}$ in $l^2(\mathbb{Z})$ $(H_{x,\omega}\phi)(n) = -b(x+(n+1)\omega)\phi(n+1) - b(x+n\omega)\phi(n-1) + a(x+n\omega)\phi(n) = E\phi(n),\ n\in\mathbb{Z},$ where $a(x),\ b(x)$ are analytic function on $\mathbb{T}$, $b$ is not identically zero, and $\omega$ obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $\rho_0=\rho_0(a,b,\omega,E_0)$, $\beta=\beta(a,b,\omega)$ such that $|L(E)-L(E')|<|E-E'|^\beta$ for any $E,E'\in (E_0-\rho_0,E_0+\rho_0)$. If $L(E)>0$ for all $E$ in some compact interval $I$ then $L(E)$ is H\"{o}lder continuous on $I$ with a H\"{o}lder exponent $\beta=\beta(a,b,\omega,I)$. In our derivation we follow the refined version of the Goldstein-Schlag method \cite{GS} developed by Bourgain and Jitomirskaya \cite{BJ}.

Localization for a Family of One-dimensional Quasiperiodic Operators of Magnetic Origin

We show strong dynamical localization for a family of one-dimensional quasiperiodic Jacobi operators of magnetic origin, throughout the regime of positive Lyapunov exponents.