On Toda lattices and orthogonal polynomials

Abstract

First, we derive a simple connection between Toda and Langmuir lattices and give a characterization of Toda lattices with the help of Stieltjes functions. Then it is shown how to generate by orthogonal polynomials in an elementary way periodic and almost periodic Toda lattices. The particles of the Toda lattice are not even restricted, as usual, to move on the real line, they may also move in the complex plane. With the help of this result, for special cases explicit solutions are obtained in terms of elliptic functions.