The Landau-Problem on the θ-Deformed Two-Torus

We study the Landau-problem on the $\theta$-deformed two-torus and use well-known projective modules to obtain perturbed spectra. For a strong magnetic field B the problem can be restricted to one particular Landau-level. First we represent generators of the algebra of the non-commutative torus as finite dimensional matrices. A second approach leads to a reducible representation with a $\theta$-dependent center. For a simple periodic potential, the rational part of the Hofstadter-butterfly spectrum is obtained.