On the criticality and the principal eigenvalue of almost periodic elliptic operators

TL;DR

This paper explores the properties of the principal eigenvalue and criticality for almost periodic elliptic operators, revealing nuanced relationships and instabilities in their spectral characteristics.

Contribution

It provides new insights into the relationship between criticality and principal eigenvalues for almost periodic elliptic operators, including counterexamples and stability analysis.

Findings

Counter-example showing criticality is not equivalent to the existence of an almost periodic principal eigenvalue

Liouville-type result established for dimension N ≤ 2

Existence of an almost periodic operator that is subcritical but admits a critical limit operator

Abstract

We review the notion and the properties of the generalised \pe\ for elliptic operators in unbounded domains, and we relate it with the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension $N\leq2$. Next, we show with a counter-example that criticality is not equivalent to the existence of an almost periodic principal eigenvalue, even for self-adjoint operators. Finally, we exhibit an almost periodic operator which is subcritical but which admits a critical limit operator. This is a manifestation of the instability character of the criticality property in the almost periodic setting.