  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.
