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

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, i.e. 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 L 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 L.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, i.e. 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 L 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 L