A hyperbolic Kac-Moody Calogero model

Abstract: A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra $AE_3$ with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincar\'e disk with a unique potential. Since the Weyl group of $AE_3$ is a $\mathbb{Z}_2$ extension of the modular group PSL(2,$\mathbb{Z}_2$), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of $AE_3$, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincar\'e series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.