Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box

TL;DR

This paper investigates the relationship between the principal eigenvalue and the landscape function in the Anderson model on large boxes, providing asymptotic analysis and partial proofs of a conjecture, with a complete proof in one dimension.

Contribution

It formulates a precise conjecture relating eigenvalues and landscape functions, offers asymptotic results, and proves the conjecture fully in one dimension.

Findings

Asymptotic behavior of the principal eigenvalue as box size increases

Partial proof of the conjecture in higher dimensions

Complete proof of the conjecture in one dimension

Abstract

We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. We give a complete proof for the one dimensional case.