Padé approximants and complex high order Toda lattices
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.
isospectral deformation of Jacobi matrix
scientific article; zbMATH DE number 1929353