Periodic Jacobi Matrices Research Articles

A useful formula for periodic Jacobi matrices on trees

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it in terms of entries of the resolvent of the matrix and its "half-tree" restrictions. This formula is closely related to the one-dimensional Thouless formula and associates a natural phase with points in the bands. This allows streamlined proofs of the gap labeling and Aomoto index theorems. We give a complete proof of gap labeling and sketch the proof of the Aomoto index theorem. We also prove a version of this formula for the Anderson model on trees.

On the number of closed gaps of discrete periodic one-dimensional operators

From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.

The new inverse eigenvalue problems for periodic and generalized periodic Jacobi matrices from their extremal spectral data

In this paper, we give sufficient conditions for the construction of periodic and generalized periodic Jacobi matrices from particular spectral data. For the construction of a periodic Jacobi matrix, the smallest and largest eigenvalues of all its leading principal submatrices, and a prescribed entry are required. For the case of the generalized periodic Jacobi matrix, an eigenvector associated with the largest eigenvalue is additionally considered. The results obtained provide algorithmic procedures that are illustrated in some numerical experiments.

Spectral resolutions for non-self-adjoint block convolution operators

The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.

A reduction algorithm for reconstructing periodic Jacobi matrices in Minkowski spaces

The periodic Jacobi inverse eigenvalue problem concerns the reconstruction of a periodic Jacobi matrix from prescribed spectral data. In Minkowski spaces, with a given signature operator H=diag(1,1,…,1,−1), the corresponding matrix is a periodic pseudo-Jacobi matrix. The inverse eigenvalue problem for such matrices consists in the reconstruction of pseudo-Jacobi matrices, with the same order and signature operator H. In this paper we solve this problem by applying Sylvester’s identity and Householder transformation. The solution number and the corresponding reconstruction algorithm are here exhibited, and illustrative numerical examples are given. Comparing this approach with the known Lanczos algorithm for reconstructing pseudo-Jacobi matrices, our method is shown to be more stable and effective.

Remarks on periodic Jacobi matrices on trees

We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well-known gap in the spectrum of the Laplacian on the upper half-plane with a hyperbolic metric. We make some conjectures about antibound states and make an interesting observation for the so-called rg-model where the underlying graph has r red and g green vertices and where any two vertices of different colors are connected by a single edge.

Periodic Jacobi matrices on trees

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.

On the inverse eigenvalue problem for periodic Jacobi matrices

ABSTRACT The problem of reconstructing a matrix with a specific structure from a partial or total spectral data is known as inverse eigenvalue problem which arises in a variety of applications. In this paper, we study a partially described inverse eigenvalue problem of periodic Jacobi matrices and prove some spectral properties of such matrices. The problem involves the reconstruction of the matrix by one eigenvalue of each of its leading principal submatrices and one eigenvector of the required matrix and one more additional piece of information. The conditions for solvability of the problem are presented and finally an algorithm and some numerical results are given.

On stability in the Borg–Hochstadt theorem for periodic Jacobi matrices

A result of Borg–Hochstadt in the theory of periodic Jacobi matrices states that such a matrix has constant diagonals as long as all gaps in its spectrum are closed (have zero length). We suggest a quantitative version of this result by proving two-sided bounds between oscillations of the matrix entries along the diagonals and the length of the maximal gap in the spectrum.

Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices

We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many contexts in pure and applied mathematics. Therefore, the study of the inverse of these matrices becomes of specific interest. However, explicit formulas for inverses are known only in a few cases, in particular when the coefficients of the diagonal entries are subjected to some restrictions. We will show that the inverse of generalized Jacobi matrices can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. In fact, recent advances in the study of linear difference equations, allow us to compute the solution of this kind of boundary value problems. So, the conditions that ensure the uniqueness of the solution of the boundary value problem leads to the invertibility conditions for the matrix, whereas that solutions for suitable problems provide explicitly the entries of the inverse matrix.

Spectral Theory, Jacobi Matrices, Continued Fractions and Difference Operators

The aim of this paper is to describe some connections between spectral theory in infinite dimensional Lie algebras, deformation theory and linearization of nonlinear dynamical systems. We explain how results from isospectral deformations, cohomology groups and algebraic geometry can be used to obtain insight into integrable systems. Another part will be dedicated to the study of infinite continued fractions and isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Also, the notion of algebraically completely integrable systems is explained and techniques to solve such systems are presented. Several nonlinear problems in mathematical physics illustrate these results.

Generalized inverse eigenvalue problems for augmented periodic Jacobi Matrices

In this paper, we propose a new method to solve the generalized inverse eigenvalue problem for periodic Jacobi matrices. Besides, we introduce a new inverse eigenvalue problem for augmented periodic Jacobi matrices, and present a corresponding stable algorithm to solve this inverse problem. Numerical experiments show that the new method we proposed is efficient.

Killip–Simon problem and Jacobi flow on GMP matrices

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert–Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuous (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is an arbitrary Jacobi matrix (of Hilbert–Schmidt class) the a.c. spectrum remains perfectly the same, that is, the interval [−2,2]. Moreover, Killip and Simon described explicitly the spectral properties for such A+B. Jointly with Damanik they generalized this result to the case of perturbations of periodic Jacobi matrices in the non-degenerated case. Recall that the spectrum of a periodic Jacobi matrix is a system of intervals of a very specific nature. Christiansen, Simon and Zinchenko posed in a review dedicated to F. Gesztesy (2013) the following question: “is there an extension of the Damanik–Killip–Simon theorem to the general finite system of intervals case?” In this paper we solve this problem completely. Our method deals with the Jacobi flow on GMP matrices. GMP matrices are probably a new object in the spectral theory. They form a certain Generalization of matrices related to the strong Moment Problem, the latter ones are a very close relative of Jacobi and CMV matrices. The Jacobi flow on them is also a probably new member of the rich family of integrable systems. Finally, related to Jacobi matrices of Killip–Simon class, analytic vector bundles and their curvature play a certain role in our construction and, at least on the level of ideology, this role is quite essential.

Lieb\u2013Thirring inequalities for complex finite gap Jacobi matrices

We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap almost periodic Jacobi matrices.

Periodic GMP Matrices

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to the class of periodic GMP matrices. In particular, we prove an analogue of the remarkable "magic formula" for this new class.

Eigenvalues for Perturbed Periodic Jacobi Matrices by the Wigner-von Neumann Approach

The Wigner-von Neumann method, which was previously used for perturbing continuous Schr\"{o}dinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary $T$-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, $\lambda$, into the operator's absolutely continuous spectrum. Introducing a new rational function, $C(\lambda;T)$, related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of $C(\lambda;T)$); in particular showing that there are only finitely many of them.

On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach

This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given.

On the measure of the spectrum of direct integrals

We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators.

Meromorphic Continuations of Finite Gap Herglotz Functions and Periodic Jacobi Matrices

We find a necessary and sufficient condition for a Herglotz function m to be the Borel transform of the spectral measure of an exponential decaying perturbation of a periodic Jacobi matrix. The condition is in terms of meromorphic continuation of m to a natural Riemann surface and the structure of its zeros and poles.The analogous result is also established for the Borel transform of the spectral measure of eventually periodic Jacobi matrices.This paper generalizes the corresponding result from the author’s (Constr Approx 36(2):267–309, 2012) for exponential perturbations of the free Jacobi matrix.

Direct and inverse spectral problems for a class of non-self-adjoint periodic tridiagonal matrices

The spectral properties of a class of tridiagonal matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The obtained results extend some results on the direct and inverse spectral problems for periodic Jacobi matrices and for some non-self-adjoint tridiagonal matrices.