Graduate Colloquium Spring 2011

April 14, 2011

Toward a theory of families of parameter-dependent commuting matrices

Haile K. Owusu (Center for Materials Theory, Department of Physics and Astronomy, Rutgers University)

We consider finite quantum integrable Hamiltonians represented as N xN real symmetric matrices linear in a coupling u and characterized by their type, M, where the corresponding number of independent commuting matrices similarly linear in a coupling is K ≡ N − M + 1. We discuss progress towards parametrizing all such Hamiltonians, including Type 1 solutions and an ansatz for the parameterization of general Type M operators. We go on to show the intimate link between these commuting Hamiltonians and the algebraic geometry of Riemann surfaces.