M.Sc Thesis
M.Sc Student Rivkind Alexandr Some Rigorous Bounds for the Anderson Model and Their Applications Department of Physics Professor Emeritus Shmuel Fishman (Deceased)

Abstract

Exploration of the nonlinear Schrödinger equation (NLSE) with a random potential motivated studies of some properties of the linear model. The main focus of the present work is on rigorous analysis of the properties of the spectrum of a finite chain with a random potential. It is shown that the spacing between eigenvalues of the discrete one dimensional Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. An explicit lower bound, given by Ce-bN, where N is the lattice size, while C and b are some finite constants is established. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications it is shown that to leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one-dimension (on the lattice) remains exponentially localized, in probability and average sense for initial conditions given by a unique eigenfunction of the linear problem. The derivative of the eigenfunctions of the linear Anderson model with respect to a potential change is bounded as well. The latter bound was refined with an improved technique which is also presented.