On the criticality and the principal eigenvalue of almost periodic elliptic operators
Luca Rossi
TL;DR
The article investigates how the generalized principal eigenvalue $\lambda_1$ for elliptic operators with almost periodic coefficients in unbounded domains relates to the Pinschover criticality framework. By defining $\lambda_1$, $\lambda_1'$, and $\lambda_1''$ and examining limit operators, it reveals Liouville-type results in low dimensions and establishes that, unlike the periodic case, criticality does not coincide with the existence of an almost periodic principal eigenfunction. The authors construct explicit counterexamples in dimension $N=1$ showing that an operator can be critical without an $\text{a.p.}$ eigenfunction and that subcritical operators can have critical limit operators, highlighting instability of the criticality notion under almost periodic perturbations. The work also surveys alternative eigenvalue notions, their interrelations, and how homogenization insights relate to $\lambda_1',\lambda_1''$ in the a.p. setting, contributing to a nuanced understanding of spectral properties in heterogeneous media.
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.