Mourre estimates for a 2D magnetic quantum Hamiltonian on strip-like domains

We consider a 2D Schrödinger operator H0 with constant magnetic field defined on a strip of finite width. The spectrum of H0 is absolutely continuous and contains a discrete set of thresholds. We perturb H0 by an electric potential V, and establish a Mourre estimate for H = H0 + V when V is periodic in the infinite direction of the strip, or decays in a suitable sense at infinity. In the periodic case, for each compact subinterval I contained in between two consecutive thresholds, we show as a corollary that the spectrum of H remains absolutely continuous in I, provided the period and the size of the perturbation are sufficiently small. In the second case we obtain that the singular continuous spectrum of H is empty, and any compact subset of the complement of the thresholds set contains at most a finite number of eigenvalues of H, each of them having finite multiplicity. Moreover these Mourre estimates together with some of their spectral consequences generalize to the case of 2D magnetic Schrödinger operators defined on R2 for suitable confining potentials modeling Dirichlet boundary conditions