Professor Simon Ruijsenaars
My research interests belong to the area of finite-dimensional integrable systems. Most of my work is focussed on N-body quantum systems of soliton type. In such systems, momenta and bindings are conserved under scattering, and the scattering is factorised as if a sequence of 2-body collisions were taking place. The pertinent quantum systems and their classical versions are at the crossroads of a great many subfields of mathematics and physics. Their nonrelativistic versions date back to the mid-seventies and bear the names of Calogero, Moser and Sutherland. In the mid-eighties, relativistic versions of these systems were discovered by myself and my then student H. Schneider at the classical level. Shortly thereafter, I found a way to quantise the systems such that their integrable character is preserved, and I introduced relativistic versions of the classical and quantum Toda systems as well.
The first step in `solvingâ€™ the relativistic systems consists in constructing/discovering joint eigenfunctions for the commuting Hamiltonians, which are analytic difference operators. The second step is to use these eigenfunctions to construct unitary Hilbert space transforms. (To be sure, neither the existence of joint eigenfunctions nor the feasibility to promote them to the kernel of a unitary transform follows from any known general results.) The sought-for transforms generalise Fourier transforms for the open Toda case and the hyperbolic version of the CMS systems, and Fourier series for the closed Toda case and the elliptic version. (The trigonometric version of the relativistic CMS systems has been solved in terms of the well-known Macdonald polynomials.)
The operators at issue have properties not shared by the well-studied classes of differential and discrete difference operators. Indeed, their symbols are exponential in the momenta, so that they do not even belong to the vast class of Fourier integral operators. As a consequence, a lot of novel problems arise in the construction of rigorous Hilbert space counterparts of the commuting analytic difference operators.
Research groups and institutes
- Applied Mathematics