Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain
| dc.creator | Foxman, JA | |
| dc.creator | Robbins, JM | |
| dc.date | 2004-11-04 | |
| dc.date | 2005-10-08 | |
| dc.date.accessioned | 2026-07-07T06:38:33Z | |
| dc.date.available | 2026-07-07T06:38:33Z | |
| dc.description | 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)^{μ(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. | |
| dc.description | 25 pages; published version | |
| dc.identifier | https://arxiv.org/abs/math-ph/0411018 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0411018 | |
| dc.identifier | Nonlinearity 18 (2005) 2795-2813 | |
| dc.identifier | doi:10.1088/0951-7715/18/6/020 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100774 | |
| dc.subject | Mathematical Physics | |
| dc.subject | 37J35; 81S10; 81S30 | |
| dc.title | Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain | |
| dc.type | text |