The band-edge behavior of the density of surfacic states

TL;DR

This paper investigates the asymptotic behavior of the surface density of states near spectral edges in a discrete Anderson model, distinguishing between fluctuating and stable edges with specific asymptotic results.

Contribution

It provides a detailed analysis of the asymptotics of surfacic state density near spectral edges, introducing new results for both fluctuating and stable edge cases.

Findings

Lifshitz tail asymptotics for fluctuating edges

Surface density of states matches constant potential case for stable edges

Asymptotic formulas derived for different edge types

Abstract

This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered : fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably "reduced" to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.