Padé approximants and complex high order Toda lattices
type: Item
instance of: scholarly article
title: Padé approximants and complex high order Toda lattices
zbMATH Open document ID: 1031.42026
DOI: 10.1016/S0377-0427(02)00866-X
published in: Journal of Computational and Applied Mathematics
publication date: 2003-06-17T00:00:00Z
review text: Given an infinite symmetric tridiagonal matrix \(J=J(t)\), the corresponding Toda lattice of order \(p\) is a system of nonlinear differential equations whose coefficients are expressed in terms of the entries of the matrix \(J^p\). For \(p=1\) Moser showed that the dynamics of the solution of the Toda equations with \(J(t)\) bounded and selfadjoint can be described in terms of the spectral measure of \(J(t)\). Later in [\textit{A. I. Aptekarev, A. Branquinho} and \textit{F. Marcellán}, J. Comput. Appl. Math. 78, 139-160 (1997; Zbl 0886.33007)] this result has been extended to \(p>1\) and \(J\) selfadjoint. Here a further generalization is achieved, since \(J\) is no longer selfadjoint: its entries are bounded but complex. Now instead of the spectral measure we must speak about the resolvent function. Its expression in continued fractions recovers the entries of \(J\), giving a procedure for determination of \(J\) and integration of the Toda equations for complex initial data.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 37K60
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 1929353
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: three-term recurrence relations
zbMATH Keywords: transformations of the measure
zbMATH Keywords: isospectral deformation of Jacobi matrix
zbMATH Keywords: direct and inverse spectral problem
zbMATH Keywords: Toda lattice
author: A. I. Aptekarev
author: A. Branquinho
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
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full work available at URL: s0377-0427(02)00866-x
OpenAlex ID: W2032644129
description: scientific article; zbMATH DE number 1929353