Lifshitz Singularity of the Integrated Density of States and Absence of Diffusion Near the Bottom of the Spectrum for A Random Hamiltonian

Abstract

Let us consider the following simple model of a stochastic Hamiltonian which describes the motion of a quantum particle interacting with a random potential: $$ {{H}_{\omega }} = - \Delta + {{\sum }_{i}}{{q}_{i}}(\omega ){{X}_{{{{C}_{i}}}}}(x) on {{L}^{2}}({{R}^{\nu }}). $$ (1) Here Ci ia a covering of Rυ by unit cubes around the sites of Zυ and qi (ω) are independent identically distributed random variables with common distribution dP(qo 0, where denotes expectation with respect to dP (qo < λ). An important role in the physics of the above model is played by the distribution of the eigenvalues of Hω N(E) defined below. We will refer in the following to N(E) as the integrated density of states. In this note we will report on some old and recent results concerning N(E) and on their application to the analysis of the diffusive behavior of the model. For general background concerning stochastic Hamiltonians we refer to [1,2] (see also Souillard in this book).