QUANTUM-MECHANICS AND QUANTUM HALL-EFFECT ON RIEMANN SURFACES

The quantum mechanics of a system of charged particles interacting wit h a magnetic field on Riemann surfaces is studied. We explicitly const ruct the wave functions of ground states in the case of a metric propo rtional to the Chem form of the theta-bundle, and the wave functions o f the Landau levels in the case of the Poincare metric. The degeneracy of the Landau levels is obtained by using the Riemann-Roch theorem. T hen we construct the Laughlin wave function on Riemann surfaces and di scuss the mathematical structure hidden in the Laughlin wave function. Moreover the degeneracy of the Laughlin states is also discussed.