Quantum mechanics on thin cylinders.

We discuss the quantum mechanics of particles of arbitrary statistics on an infinite cylinder with and without a magnetic field perpendicular to the surface. In the presence of a magnetic field, the translational symmetry along the cylinder is broken down to a discrete one by the Aharonov-Bohm effect. For interacting fermion in a strong field we get an effective one-dimensional lattice model that in a limit can be mapped on an Ising chain. We also show that a system of anyons on a cylinder are, in a certain limit, closely related to the one-dimensional (integrable) Sutherland model. By order of magnitude estimates we demonstrate that none of these effects are likely to be experimentally observed with present techniques