Speaker
Gage Hoefer, Dartmouth College
Abstract
A fundamental question in control theory is (as the name suggests) controllability of a given system: can one reach a particular state from an arbitrary initial state by manipulating a set of controls (often vector fields or Hamiltonians, depending on the setting) which govern the dynamics of the underlying system? Much is known about controllability for quantum systems modeled on finite-dimensional Hilbert spaces, and a particularly useful framework for answering such questions is the use of Lie theory for Lie subalgebras generated by the given Hamiltonians. In the infinite-dimensional setting, however, numerous problems arise due to the high likelihood of a system possessing unbounded Hamiltonians with complicated spectral information. In this talk, we will propose a setup which helps avoid the difficulties one encounters dealing with unbounded operators, by requiring all controls to share a specific relation— known as affiliation— with a von Neumann algebra M acting on the same Hilbert space. We show how we may recover analogues of results from the finite-dimensional setting under these assumptions, using the theory of von Neumann algebras and their type classification. We also discuss approximation results which preserve affiliation relations for controls which possess challenging spectral properties, and how this approach can be applied to the study of classical dynamical systems through the Koopman operator formalism.
This talk is based on work-in-progress with Dimitrios Giannakis.