Random SchrÜdinger operators of Anderson type with generalized Laplacians and sparse potentials

• Main Author: Poulin, Philippe.
• Format: Others
• Language: en
• Published: McGill University 2006
• Subjects: Mathematics.
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=102719

The first part of the thesis concerns Green's functions of discrete Laplacians on lattices. In the continuous case, it is well known that the corresponding Green's functions decay polynomially. However, an identical proof of this fact fails in the discrete case, since the constant energy surfaces of the discrete Laplacian are not convex. Two approaches are presented to turn around this problem. One consists of adapting the stationary phase method in order to treat non convex surfaces admitting kappa > 0 non vanishing principal curvatures at each point; as suggested by Littman. The other consists of changing the discretization of the Laplacian, as suggested by Molchanov and Vainberg. === The second part of the thesis concerns random Schrodinger operators of type Anderson on the d-dimensional lattice. Sufficient conditions are presented for such operators, H = Delta + V, to satisfy almost surely the following, remarkable spectral and scattering properties: (1) Outside spec(Delta), the spectrum of H is pure point with exponentially decaying eigenfunctions (so-called Anderson localization). Examples where the spectrum of H is equal to the whole real line are also exhibited, in which case the eigenvalues of H are in addition dense in R \spec(Delta); (2) Inside spec(Delta), the spectrum of H is purely absolutely continuous (so-called delocalization); (3) Inside spec(Delta), the wave operators between H and Delta exist and are complete. Such Anderson operators are exhibited for the first time in the literature. Using the estimate of the first, part of the thesis, the mentioned sufficient conditions appear to be sparseness conditions on the support of the potential.