An Infinite-Dimensional Hamiltonian System on Projective Hilbert Space

Abstract

We consider here the explicit integration of a Hamiltonian system on infinite-dimensional complex projective space. The Hamiltonian, which is the restriction of a linear functional to this projective space, arises in the problem of line fitting in complex Hilbert space (or, equivalently, the problem of functional approximation) or as the expectation value of a model quantum mechanical system. We formulate the system here as a Lax system with parameter, showing how this leads to an infinite set of conserved integrals associated with the problem and to an explicit formulation of the flow in action-angle form via an extension of some work of J. Moser. In addition, we find the algebraic curve naturally associated with the system.