Related papers: The Ten Martini Problem

We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and…

The Clifford spectrum is a form of joint spectrum for noncommuting matrices. This theory has been applied in photonics, condensed matter and string theory. In applications, the Clifford spectrum can be efficiently approximated using…

We give self-contained presentation of results related to the Kadison-Singer problem, which was recently solved by Marcus, Spielman, and Srivastava. This problem connects with unusually large number of areas including: operator algebras…

We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field…

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of…

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…

For $\xi \in \big( 0, \frac{1}{2} \big)$, let $E_{\xi}$ be the perfect symmetric set associated with $\xi$, that is $$E_{\xi} = \Big\{ \exp \Big( 2i \pi (1-\xi) \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \Big) : \, \epsilon_{n} = 0…

In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2--spectral set for the matrix A, we propose a study of various constants. We review some partial results, many problems are still open. We describe our…

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for $\Sigma^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\mathcal{U}$…

Using standard techniques from combinatorics, model theory, and algebraic geometry, we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns. Also, we point out a counterexample to a…

We study Schr\"odinger operators on $\R$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral…

We prove a HVZ theorem for a general class of no-pair Hamiltonians describing an atom or positively charged ion with several electrons in the presence of a classical external magnetic field. Moreover, we show that there exist infinitely…

Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all…

We numerically study energy spectra and localization properties of the double exchange model at irrational filling factor. To obtain variational ground state, we use a mumerical technique in momentum space by ``embedded'' boundary condition…

We develop the spectral and scattering theory for self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We construct…

We prove that any spectrum is equivalent to the nonconnective K-theory of a stable $\infty$-category. We use these results to construct a stable $\infty$-category $\mathcal{C}$ with a bounded t-structure such that…

Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $\psi$ let $ W_{\cal A}(\psi)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely…

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance…

We consider quantum systems consisting of a linear chain of n harmonic oscillators coupled by a nearest neighbour interaction of the form $-q_r q_{r+1}$ ($q_r$ refers to the position of the $r$th oscillator). In principle, such systems are…

Random Matrix Theory (RMT) is a powerful statistical tool to model spectral fluctuations. This approach has also found fruitful application in Quantum Chromodynamics (QCD). Importantly, RMT provides very efficient means to separate…