A Matrix-Valued Point Interactions Model

We study a matrix-valued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that, away from a discrete set, its Lyapunov exponents do not vanish. For this we use a criterion by Gol’dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its Hölder continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.