Related papers: Continuity of spectral averaging
The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schr\"odinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was…
For any regular cardinal $\kappa$ and ordinal $\eta<\kappa^{++}$ it is consistent that $2^{\kappa}$ is as large as you wish, and every function $f:\eta \to [\kappa,2^{\kappa}]\cap Card$ with $f(\alpha)=\kappa$ for $cf(\alpha)<\kappa$ is the…
We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a…
We show that the Continuum Hypothesis implies that for every $0<d_1\leq d_2<n$ the measure spaces $(\RR^n,\iM_{\iH^{d_1}},\iH^{d_1})$ and $(\RR^n,\iM_{\iH^{d_2}},\iH^{d_2})$ are isomorphic, where $\iH^d$ is $d$-dimensional Hausdorff measure…
In this survey paper we review classical results and recent progress about a certain topic in the spectral theory of two-dimensional canonical systems. Namely, we consider the questions whether the spectrum $\sigma$ is discrete, and if it…
Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss…
It is well-established that the spectral measure for one-frequency Schr\"odinger operators with Diophantine frequencies exhibits optimal $1/2$-H\"older continuity within the absolutely continuous spectrum. This study extends these findings…
In the present paper we sketch the proof of the fact that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $0<\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic…
In this paper, we will show the Yau's gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $\kappa$, $\kappa\geq0$, in the sense of Alexandrov. As a direct application, it gives some…
Let $G$ be a locally compact group and $\mu$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(\pi, \cal H)$ of $G,$ we study spectral properties of…
In this paper, we investigate the relationship between frame bounds and spectral gaps. By introducing the notion of \emph{essential minimum(maximal) spectral gap}, we provide a local characterization of Landau's theorem \cite{Lan67}. As an…
A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. Averaging over boundary conditions leads to explicit formulas for the averaged spectral measure which can…
The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap of finite dimensional systems is given by Theorem 1.1, in terms…
The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case…
We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches…
For convex and sequential effect algebras, we study spectrality in the sense of Foulis. We show that under additional conditions (strong archimedeanity, closedness in norm and a certain monotonicity property of the sequential product), such…
The aim of this paper is to extend the notion of the spectral order for finite families of pairwise commuting bounded and unbounded selfadjoint operators in Hilbert space. It is shown that the multidimensional spectral order $\preccurlyeq$…
In a recent work, Klartag gave an improved version of Lichnerowicz' spectral gap bound for uniformly log-concave measures, which improves on the classical estimate by taking into account the covariance matrix. We analyze the equality cases…
We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let $s$ be the Hausdorff dimension of the spectrum. For $V>20$, we show that the restriction of the…
The conformal loop ensemble $\mathrm{CLE}_{\kappa}$ is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter $\kappa$ varies between $8/3$ and…