Related papers: A Spectral Equivalence for Jacobi Matrices
Jacobi matrices probably are the most classical object in spectral theory, while CMV matrices are a comparably fresh one, although they are related to a very classical topic, namely to orhtogonal polynomials on the unit circle (in the same…
We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition.
We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it. This allows new, streamlined proofs of the gap labeling and Aomoto index theorems. We prove a version of this new…
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…
We study how the spectrum of a Jacobi operator changes when this operator is modified by a certain finite rank perturbation. The operator corresponds to an infinite mass-spring system and the perturbation is obtained by modifying one…
Representation of analytic functions as convergent series in Jacobi polynomials $P_n^{(a,b)}$ is reformulated using a unified approach for almost all complex $a, b$. The coefficients of the series are given as usual integrals in the…
By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the…
We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be…
We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the…
The paper studies the global convergence of the block Jacobi me\-thod for symmetric matrices. Given a symmetric matrix $A$ of order $n$, the method generates a sequence of matrices by the rule $A^{(k+1)}=U_k^TA^{(k)}U_k$, $k\geq0$, where…
We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the…
We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi…
In a recent paper a class of infinite Jacobi matrices with discrete character of spectra has been introduced. With each Jacobi matrix from this class an analytic function is associated, called the characteristic function, whose zero set…
We give a complete solution of the scattering problem for Jacobi matrices from a class which was recently introduced by E. Ryckman. We characterize the scattering data for this class and illustrate the inverse scattering on some simple…
We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The…
Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq…
We show that Jacobi fields along harmonic maps between suitable spaces preserve conformality, holomorphicity, real isotropy and complex isotropy to first order; this last being one of the key tools in the proof by Lemaire and the author of…
Let $\fre\subset\bbR$ be a finite union of $\ell+1$ disjoint closed intervals and denote by $\omega_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\fre$ is the set of all integral combinations of $\omega_1,...,…
In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of…
We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph…