Direct and inverse spectral transform for the relativistic Toda lattice and theconnection with Laurent orthogonal polynomials

Abstract

We introduce a spectral transform for the finite relativistic Toda lattice (RTL) ingeneralized form. In the nonrelativistic case, Moser constructed a spectraltransform from the spectral theory of symmetric Jacobi matrices. Here we use anon-symmetric generalized eigenvalue problem for a pair of bidiagonal matrices(L, M)to define the spectral transform for the RTL. The inversespectral transform is described in terms of a terminatingT-fraction.The generalized eigenvalues are constants of motion and the auxiliary spectraldata have explicit time evolution. Using the connection with the theory of Laurentorthogonal polynomials, we study the long-time behaviour of the RTL. As in thecase of the Toda lattice the matrix entries have asymptotic limits. We show thatL tends toan upper Hessenberg matrix with the generalized eigenvalues sorted on the diagonal,while Mtends to the identity matrix.