Open Collections

Abstract

This thesis is divided as follows: The first chapter introduces the main ideas intuitively while assuming an acquaintance with quantum mechanics. The second chapter exposes the mathematical setting of the problem under investigation and contains a brief excursion into self-adjointness of unbounded operators. The core of the thesis is contained in the third chapter where a conjugate operator A is defined in order that the Mourre estimate for the discrete Hamiltonian H = L + Q is shown to hold for a binary tree configuration space, where L is the discrete Laplacian and Q a discrete potential either of short-range type or of long-range type satisfying a first order difference condition. In the last chapter the result from chapter three as well as some extra material is used to show that asymptotic completeness holds for a one-body system having a binary tree configuration space with either of the following three types of potential: 1) long-range potentials satisfying first and second order difference conditions 2) short-range potentials of order o(|v|⁻¹) and satisfying a second order difference condition or 3) short-range potentials of order o(|v|⁻²).