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

We review the notion and the properties of the generalized principal eigenvalue for elliptic operators in unbounded domains, and we relate it to the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension N ≤ 2. Next, we show with a counterexample that criticality is not equivalent to the existence of an almost periodic principal eigenfunction, 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.