Some Transport and Spectral Properties of Disordered Media

Abstract

In this paper, we study some relations between the local geometry of a potential V on R ⁿ and the spectrum of the corresponding self-adjoint Hamiltonian H = −Δ + V on L ²(R ⁿ). More specifically, for a fixed energy interval I,we want to examine the effects of strong local fluctuations of V near energies in I on the spectrum of H, σ(H), in the interval I. We consider, firstly, the dynamics generated by such a potential as expressed by the diffusion constant. We give sufficient conditions on V in terms of the potential barriers of V at energies in I so that the system evolves in a sub-diffusive manner. Secondly, we study the nature of the spectrum of H in I. We show that with sufficiently strong local fluctuations of V the absolutely continuous spectrum of H in I is empty. Our motivation for these preliminary studies is to explore these questions in a simple disordered but deterministic situation. We are primarily interested in one-electron models of random disordered media. Here, we give an application of these ideas to the one-dimensional Poisson model and prove the absence of absolutely continuous spectrum at all energies (when the single-site potential u ≥ 0) and the absence of diffusion at low energies. We will discuss the Anderson-type and Poisson-type models for potentials in R ⁿ in another paper.

supported in part by NSF grants DMS91-06479 and INT 90-15895