Related papers: Jacobi matrices with lacunary spectrum
Based on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka-Volterra lattice.…
We study the double scaling limit for unitary invariant ensembles of random matrices with non analytic potentials and find the asymptotic expansion for the entries of the corresponding Jacobi matrix. Our approach is based on the…
Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…
We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when $0$ lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when the orthogonality measure is absolutely…
We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the…
We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function…
In this paper we study spectral properties of Jacobi operators. In particular, we prove two main results: (1) that perturbing the diagonal coefficients of Jacobi operator, in an appropriate sense, results in exponential localization, and…
We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to…
We use a classical result of Gollinski and Ibragimov to prove an analog of the strong Szego theorem for Jacobi matrices on $l^2(\N)$. In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences…
We show that many existing divisibility sequences can be seen as sequences of determinants of matrix divisibility sequences, which arise naturally as Jacobian matrices associated to groups of maps on affine spaces.
A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the distance matrix, distance Laplacian, and distance signless Laplacian, in…
Consider an ordinary differential equation which has a Lax pair representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such an equation can be…
We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Various parametric forms of trajectories are associated with different integrals…
We consider Jacobi matrices and Schrodinger operators that are reflectionless on an interval. We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data, that is due to Marchenko. Then…
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
In this paper, we compute the spectral norms of the matrices related with integer squences and we give two examples related with Fibonacci and Lucas numbers.
We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we study the topological structure of the…
We compute the Laplacian spectra of singular area-minimising hypersurfaces in the hyperbolic space with prescribed asymptotic data. We also obtain similar results in higher codimension, and explore related extremal properties of the bottom…
We construct so called Darboux matrices and fundamental solutions in the important case of the generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in…
We study several related aspects of reflectionless Jacobi matrices. Our first set of results deals with the singular part of reflectionless measures. We then introduce and discuss Lyapunov exponents, density of states measures, and other…