On the extension of singular linear infinite-dimensional Hamiltonian flows

Abstract

We study the phenomenon of phase trajectories of a Hamiltonian system going to infinity in a finite time, the phase space of which is a separable Hilbert space. It is shown that if the Hamiltonian is a densely defined quadratic form on the phase space, which is not majorized either from below or from above by the quadratic form of the Hilbert norm, then the phase trajectories allow going to infinity in a finite time. To describe the phase flow of such Hamiltonian systems, an extended phase space is introduced, which is a locally convex space to which the Hamiltonian function, trajectories of the Hamiltonian system, and the symplectic form defined on the original Hilbert space can be extended. Flowinvariant measures on extended space are also studied. The properties of the Koopman unitary representation of the extended phase flow in the Hilbert space of functions that are quadratically integrable with respect to an invariant measure are investigated.