Alternative structures and bi-Hamiltonian systems on a Hilbert space

TL;DR

This paper explores bi-unitary transformations in quantum systems on Hilbert spaces, introducing Hermitian structures in generic relative position, and extends finite-dimensional bi-Hamiltonian results to infinite-dimensional settings.

Contribution

It introduces the concept of Hermitian structures in generic relative position and generalizes finite-dimensional bi-Hamiltonian system results to infinite-dimensional Hilbert spaces.

Findings

Characterization of bi-unitary transformations in infinite dimensions

Conditions for Hermitian structures to be in generic relative position

Extension of finite-dimensional bi-Hamiltonian results to Hilbert spaces

Abstract

We discuss transformations generated by dynamical quantum systems which are bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an infinite-dimensional complex Hilbert space. We introduce the notion of Hermitian structures in generic relative position. We provide few necessary and sufficient conditions for two Hermitian structures to be in generic relative position to better illustrate the relevance of this notion. The group of bi-unitary transformations is considered in both the generic and non-generic case. Finally, we generalize the analysis to real Hilbert spaces and extend to infinite dimensions results already available in the framework of finite-dimensional linear bi-Hamiltonian systems.