Analysis of Exploding Solutions of an Infinite-Dimensional Linear Hamiltonian System in Phase Space Extensions

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. Invariant measures are used to study linear Hamiltonian systems whose trajectories go to infinity in the phase space in a finite time. An extension of the phase space is defined and extensions of the symplectic form and the Hamiltonian function to the extended space are given. The trajectories of the extended Hamiltonian system determine extensions of solutions of the original Hamiltonian equations through the moment when they go to infinity. The Hamiltonian flow in the extended phase space admits a unitary representation using the Koopman group. Subspaces of strong continuity of the unitary representation are found and properties of its generator are described.