On the extension of singular linear infinite-dimensional Hamiltonian flows

TL;DR

This paper explores Hamiltonian flows in infinite-dimensional Hilbert spaces, focusing on invariant measures and their application to linear systems with unbounded energy growth, extending the understanding of singular solutions and phase space dynamics.

Contribution

It introduces a method to describe Hamiltonian flows using invariant measures and unitary groups, extending the analysis to systems with singularities and unbounded energy increase.

Findings

Invariant measures facilitate the description of Hamiltonian flows in infinite dimensions.

Solutions with singularities can be characterized via phase flow and Koopman representation.

The approach applies to linear systems exhibiting unlimited kinetic energy growth.

Abstract

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the flows of completely integrable Hamiltonian systems are investigated. These construction gives the opportunity to describe Hamiltonian flows in the phase space by means of unitary groups in the space of functions that are quadratically integrable by the invariant measure. Invariant measures are applied to the study of model linear Hamiltonian systems that admit features of the type of unlimited increase in kinetic energy over a finite time. Due to this approach solutions of Hamilton equations that admit singularities can be described by means of the phase flow in the extended phase space and by the corresponding Koopman representation of the unitary