Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

TL;DR

This paper analyzes the singularities and Maslov indices of the periodic Toda chain, revealing their geometric structure, spectral degeneracies, and implications for the system's stability and topology.

Contribution

It establishes a correspondence between singularities and degenerate eigenvalues of Lax matrices, characterizes their nondegeneracy and elliptic nature, and relates Maslov indices to spectral holonomies.

Findings

Singularities coincide with degenerate eigenvalues of Lax matrices.

Singularities are nondegenerate and form symplectic submanifolds.

Maslov index relates to spectral data and eigenvector bundle holonomies.

Abstract

The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear relations amongst the gradients of the integrals of motion, coincide with points where there are k (doubly) degenerate eigenvalues of representatives L and Lbar of the two inequivalent classes of Lax matrices (corresponding to degenerate periodic or antiperiodic solutions of the associated second-order difference equation). The singularities are shown to be nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold. Sigma_k is shown to be of elliptic type, and the frequencies of transverse oscillations under Hamiltonians which fix Sigma_k are computed in terms of spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is given by the product of the holonomies (equal to +/- 1) of the even- (or odd-) indexed eigenvector bundles of L and Lmat.