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### Abstract

In this PhD thesis we deal with two mathematical problems arising from quantum mechanics. We consider a spinless non relativistic quantum particle whose configuration space is a two dimensional surface S. We also suppose that the particle feels the effect of an homogeneous magnetic field perpendicular to the surface S. In the first case S = R × SL1, the infinite cylinder of circumference L, corresponding to periodic boundary conditions, while in the second one S = R2. In both cases the particle feels the effect of an additional suitable potential. We are thus left with the study of two specific classes of Schrödinger operators. The operator of the first class generates the dynamics of the particle when it is submitted to an Anderson-type random potential, as well as to a non random potential confining the particle along the cylinder axis in an interval of length L. In this case we describe the spectrum and classify it by the quantum mechanical current carried by the corresponding eigenfunctions. We prove that there are spectral regions in which all the eigenvalues have an order one current with respect to L, and spectral regions where eigenvalues with order one current and eigenvalues with infinitesimal current with respect to L are intermixed. These results are relevant for the theory of the integer quantum Hall effect. The second Schrödinger operator class corresponds to the physical situation where the potential is the sum of a "local" potential and of a potential due to a weak constant electric field F. In this case we show that the resonant states, induced by the electric field, decay exponentially at a rate given by the imaginary part of the eigenvalues of some non self-adjoint operator. Moreover we prove an upper bound on this imaginary part that turns out to be of order exp(-1/F2) as F goes to zero. Therefore the lifetime of the resonant states is at least of order exp(-1/F2).