Integrable Systems and Topology of Isospectral Manifolds

TL;DR

This paper explores the complex topology of isospectral manifolds in integrable systems, especially where singularities prevent direct application of classical theorems, and introduces new results using the Volterra system.

Contribution

It extends the understanding of isospectral manifold topology beyond classical cases by analyzing zero-diagonal Jacobi matrices with the Volterra system.

Findings

Topology of isospectral varieties of zero-diagonal Jacobi matrices is characterized.

Flow methods can analyze topology even with singularities in the phase space.

New results generalize previous studies on Toda lattice and related systems.

Abstract

The well known Liouville-Arnold theorem says that if a level surface of integrals of an integrable system is compact and connected, then it is a torus. However, in some important examples of integrable systems the topology of a level surface of integrals is quite complicated. This is due to the fact that in these examples the phase space has points where either the Hamiltonian is singular or the symplectic form is singular or degenerate. In such situations the Liouville-Arnold theorem does not apply. However, sometimes it is possible to define the corresponding flow on the whole level surface of integrals and use this flow to investigate the topology. Tomei (1982) and Fried (1986) used the Toda lattice to study the topology of the isospectral variety of Jacobi matrices. We recall these results and we also expose new results concerning the topology of the isospectral variety of zero-diagonal Jacobi matrices. This topology is studied using the Volterra system.