Related papers: Spectra self-similarity for almost Mathieu operato…

We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu…

We prove that for Diophantine \om and almost every \th, the almost Mathieu operator, (H_{\omega,\lambda,\theta}\Psi)(n)=\Psi(n+1) + \Psi(n-1) + \lambda\cos 2\pi(\omega n +\theta)\Psi(n), exhibits localization for \lambda > 2 and purely…

We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space.…

We show that the position operator for a class of $f$-deformed oscillators has a fractal spectrum, homeomorphic to the Cantor set, via a unitary transformation to Harper's model. The class corresponds to a choice of ergodic operators for…

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of…

We consider the variety of spectral measures that are induced by quasitraces on the spectrum of a self-adjoint operator in a simple separable unital and Z-stable C$^*$-algebra. This amounts to a continuous map from the simplex of…

A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy…

We study the spectral location of strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide"…

Let M_n be the collection of n x n complex matrices equipped with operator norm. Suppose U, V \in M_n are two unitary matrices, each possessing a gap larger than \Delta in their spectrum, which satisfy ||UV-VU|| \le \epsilon. Then it is…

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical…

We point out that the recent conjectural solution to the spectral problem for the Hamiltonian $H=e^{x}+e^{-x}+e^{p}+e^{-p}$ in terms of the refined topological invariants of a local Calabi-Yau geometry has an intimate relation with…

We consider the spectrum of the almost Mathieu operator $H_\alpha$ with frequency $\alpha$ and in the case of the critical coupling. Let an irrational $\alpha$ be such that $|\alpha-p_n/q_n|<c q_n^{-\varkappa}$, where $p_n/q_n$,…

This paper focuses on the fractal characteristics of the absolutely continuous spectral measure of the subcritical almost Mathieu operator (AMO) and Diophantine frequency. In particular, we give a complete description of the (classical)…

We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by showing that approximate symmetry operators---unitary operators whose commutators with the Hamiltonian…

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling…

The spectral analysis of discretized one-dimensional Schr\"{o}dinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is…

We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schr\"odinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu…

We estimate the norm of the almost Mathieu operator $H_{\theta,\lambda} =U+U^*+(\lambda /2)(V+V^*)$ in the rotation $C^*$-algebra $A_\theta =C^*(U,V unitaries;UV=e^{2\pi i\theta} VU)$. In this process, we significantly improve the…

In this work the spectral theory of self-adjoint operator $A$ represented by Jacobi matrix is considered. The approach is based on the continued fraction representation of the resolvent matrix element of $A$. Different criteria of absolute…

Let theta = p/q with p and q relatively prime and u and v a pair of unitaries such that u v = e^{i theta} v u, where u and v generate the rotation C*-algebra A_theta. Let h_{theta, lambda} = u + u^{-1} + lambda/2(v + v^{-1}) be the almost…