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

Abstract

We introduce a spectral transform for the finite relativistic Toda lattice (RTL) in generalized form. In the nonrelativistic case, Moser constructed a spectral transform from the spectral theory of symmetric Jacobi matrices. Here we use a non-symmetric generalized eigenvalue problem for a pair of bidiagonal matrices (L,M) to define the spectral transform for the RTL. The inverse spectral transform is described in terms of a terminating T-fraction. The generalized eigenvalues are constants of motion and the auxiliary spectral data have explicit time evolution. Using the connection with the theory of Laurent orthogonal polynomials, we study the long-time behaviour of the RTL. As in the case of the Toda lattice the matrix entries have asymptotic limits. We show that L tends to an upper Hessenberg matrix with the generalized eigenvalues sorted on the diagonal, while M tends to the identity matrix.

24 pages, 9 figures

References in corpus (3)

• New integrable systems related to the relativistic Toda lattice
• A discrete time relativistic Toda lattice
• Faces of Relativistic Toda Chain

Cited by in corpus (5)

• Multidimensional Toda Lattices: Continuous and Discrete Time
• The Jacobi matrices approach to Nevanlinna-Pick problems
• Construction of the solution of the inverse spectral problem for a system depending rationally on the spectral parameter, Borg-Marchenko-type theorem, and sine-Gordon equation
• Spectral transformation associated with a perturbed $R_I$ type recurrence relation
• Lax matrices for a 1-parameter subfamily of van Diejen--Toda chains