Complex Jacobi Matrices Research Articles

Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to RII-recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices.

Spectral Properties of Some Complex Jacobi Matrices

We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide uniform asymptotics of their generalised eigenvectors. We illustrate our results by considering complex perturbations of real Jacobi matrices belonging to several classes: asymptotically periodic, periodically modulated and the blend of these two. Moreover, we provide conditions implying existence of a unique closed extension. The method of the proof is based on the analysis of a generalisation of shifted Turán determinants to the complex setting.

The Characteristic Function for Complex Doubly Infinite Jacobi Matrices

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.

On extremal properties of Jacobian elliptic functions with complex modulus

A thorough analysis of values of the function m↦sn(K(m)u|m) for complex parameter m and u∈(0,1) is given. First, it is proved that the absolute value of this function never exceeds 1 if m does not belong to the region in C determined by inequalities |z−1|<1 and |z|>1. The global maximum of the function under investigation is shown to be always located in this region. More precisely, it is proved that if u≤1/2, then the global maximum is located at m=1 with the value equal to 1. While if u>1/2, then the global maximum is located in the interval (1,2) and its value exceeds 1. In addition, more subtle extremal properties are studied numerically. Finally, applications in a Laplace-type integral and spectral analysis of some complex Jacobi matrices are presented.

Asymptotics of the discrete spectrum for complex Jacobi matrices

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l 2 (N):

RETRACTED On construction of a complex finite Jacobi matrix from two spectra

This paper concerns with the inverse spectral problem for two spectra of finite order complex Jacobi matrices (tri-diagonal symmetric matrices with complex entries). The problem is to reconstruct the matrix using two sets of eigenvalues, one for the original Jacobi matrix and one for the matrix obtained by replacing the last diagonal element of the Jacobi matrix by some other number. The uniqueness and existence results for solution of the inverse problem are established and an explicit procedure of reconstruction of the matrix from the two spectra is given.

Solution of the finite complex Toda lattice by the method of inverse spectral problem

We show that the finite Toda lattice with complex-valued initial data can be integrated by the method of inverse spectral problem. For this goal spectral data for complex Jacobi matrices are introduced and an inverse spectral problem with respect to the spectral data is solved. The time evolution of the spectral data for the Jacobi matrix associated with the solution of the Toda lattice is computed. Using the solution of the inverse spectral problem with respect to the time-dependent spectral data we reconstruct the time-dependent Jacobi matrix and hence the desired solution of the finite complex Toda lattice.

The linear pencil approach to rational interpolation

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils z B − A of two tridiagonal matrices A , B , following Spiridonov and Zhedanov. In the present paper, as well as revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Padé approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to ∞ , we compute the spectrum and the numerical range of the underlying linear pencil explicitly.

An inverse spectral problem for complex Jacobi matrices

We introduce the concept of generalized spectral function for finite order complex Jacobi matrices and solve the inverse problem with respect to the generalized spectral function. The results obtained can be used for solving of initial-boundary value problems for finite nonlinear Toda lattices with the complex-valued initial conditions by means of the inverse spectral problem method.

A Blaschke-type condition and its application to complex Jacobi matrices

We obtain a Blaschke-type necessary conditions on zeros of analytic functions on the unit disk with different types of exponential growth at the boundary. These conditions are used to prove Lieb-Thirring-type inequalities for the eigenvalues of complex Jacobi matrices.

Lieb–Thirring Bounds for Complex Jacobi Matrices

We obtain various versions of classical Lieb–Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan–Mirski Lemma and seems to be fairly general.

Discrete spectrum for complex perturbations of periodic Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices, which are compact perturbations of real periodic Jacobi matrix. The condition sufficient for the lack of the discrete spectrum for such matrices is given.

Limit sets for the discrete spectrum of complex Jacobi matrices

The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete Laplacian is studied. The precise stabilization rate (in the sense of order) of the matrix elements ensuring the finiteness of the discrete spectrum is found. An example of a Jacobi matrix with discrete spectrum having a unique limit point is constructed. These results are discrete analogues of Pavlov's well-known results on Schrödinger operators with complex potential on a half-axis.

On the location of the discrete spectrum for complex Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices that are compact perturbations of the discrete Laplacian. The condition sufficient for the lack of a discrete spectrum for such matrices is given.

On the determinacy of complex Jacobi matrices

The aim of this paper is to explore to which extend the theory of the Hamburger moment problem for real Jacobi matrices generalizes to the case of complex Jacobi matrices. In particular, we characterize the indeterminacy in terms of uniqueness of closed extensions of Jacobi matrices, and discuss the link to the growth of the smallest singular values of the underlying Hankel matrices. As a byproduct, we give a positive answer to the open question whether determinacy is preserved under bounded perturbations.

Complex Jacobi matrices and quadrature rules

Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = ?n+1pn+1(x) + ?npn(x) + ?npn-1(x), p-1(x)=0 po(x)=1 with ?n ? 0, n ? N, ?0 = 1, an infinite Jacobi matrix can be associated in the following way ? ? ??0 ?1 0...? ? ? ??1 ?1 ?2...? ? ? J = ?0 ?2 ?2...? ?.... ? ?.... ? ?....? ? ? In the general case if the sequences {?n} or {?n} are complex the associated Jacobi matrix is complex. Under the condition that both sequences {?n}and {?n} are uniformly bounded, the associated Jacobi matrix can be understood as a linear operator J acting on ?2, the space of all complex square-summable sequences, where the value of the operator J at the vector x is a product of an infinite vector x and an infinite matrix J in the matrix sense. The case when the sequences {?n} and {?n} are not uniformly bounded, an operator acting on ?2 can not be defined that easily. Additional properties of the sequence of orthogonal polynomials are needed in order to be able to define the operator uniquely. The case when the sequences ?n and ?n are real is very well understood. The spectra of the Jacobi matrix J equals the support of the measure of orthogonality for the given sequence of orthogonal polynomials. All zeros of orthogonal polynomials are real, simple and interlace, contained in the convex hull of the spectra of the Jacobi operator associated with the infinite Jacobi matrix J. Every point in ?(J) attracts zeros of orthogonal polynomials. An application of orthogonal polynomials is the construction of quadrature rules for the approximation of integration with respect to the measure of orthogonality. For arbitrary sequences {?n} and {?n} the situation is changed dramatically. Zeros of orthogonal polynomials need not be simple; they are not real and they do not necessarily lie in the convex hull of ?(J). There is also a little known about convergence results of related quadrature rule. Only in recent years a connection between complex Jacobi matrices and related orthogonal polynomials is interesting again (see [2]). Studies of complex Jacobi operators should lead to a better understanding of related orthogonal polynomials, but also the study of orthogonal polynomials with the complex Jacobi matrices should put more light on the non-hermitian banded symmetric matrices. In this lecture some results are given about complex Jacobi matrices and related quadrature rules, and also some interesting examples are presented.

Complex Jacobi matrices

Complex Jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials (FOPs). The latter are essential tools in several fields of numerical analysis, for instance in the context of iterative methods for solving large systems of linear equations, or in the study of Padé approximation and Jacobi continued fractions. In this paper we present some known and some new results on FOPs in terms of spectral properties of the underlying (infinite) Jacobi matrix, with a special emphasis to unbounded recurrence coefficients. Here we recover several classical results for real Jacobi matrices. The inverse problem of characterizing properties of the Jacobi operator in terms of FOPs and other solutions of a given three-term recurrence is also investigated. This enables us to give results on the approximation of the resolvent by inverses of finite sections, with applications to the convergence of Padé approximants.