Scattering on the system of the sparse bumps: multidimensional case

Abstract

The paper contains analysis of the spectrum of multidimensional Schrodinger operators on the lattice with sparse potentials. The latter means that the potential is an infinite sum of “bumps” with distances between their supports going to infinity. It is proven that the operator has an absolutely continuous spectrum on the interval [—2d, 2d] and it typically has a pure point spectrum outside of this interval. The pure point spectrum is usually dense on a set of positive measure, and the support of p.p. spectrum is described. Such a coexistence of two different spectral components can be considered as a justification (for our model) of Anderson's “mobility edges” conjecture.

Keywords:

• Schrödinger operator
• sparse potential
• Anderson conjecture