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

Technical Summary: On the criticality and the principal eigenvalue of almost periodic elliptic operators

Problem Statement
The paper investigates the relationship between the criticality theory and the notion of a generalised principal eigenvalue for elliptic operators defined on unbounded domains, specifically focusing on operators with almost periodic (a.p.) coefficients. While the principal eigenvalue is classically well-defined for operators on bounded domains or periodic operators (via the Krein-Rutman theory), its definition and properties become ambiguous in the almost periodic setting. The central problem is to determine whether the existence of an a.p. principal eigenvalue (an eigenvalue admitting an a.p. positive eigenfunction) is equivalent to the operator being critical (possessing a one-dimensional space of positive solutions). Furthermore, the paper examines the stability of criticality under the passage to limit operators in the almost periodic context.

Methodology
The author employs a combination of spectral theory, elliptic PDE analysis, and constructive counter-examples.

1. Definitions and Framework: The paper utilizes the framework of generalised principal eigenvalues ($\lambda_1, \lambda'_1, \lambda''_1$) introduced by Berestycki, Hamel, Nirenberg, and Varadhan for unbounded domains. It distinguishes between supercritical, critical, and subcritical operators based on the dimension of the space of positive solutions to $Lu=0$ or the existence of positive supersolutions.
2. Liouville-type Results: The analysis leverages Liouville-type theorems (specifically those by Nordmann and Pinchover) to establish conditions under which the existence of a bounded positive solution implies criticality in low dimensions ($N \le 2$).
3. Constructive Counter-examples: The core of the methodology involves the explicit construction of almost periodic functions and operators to disprove potential equivalences. The author utilizes a specific function $b(x)$, constructed in a previous work [27], which is limit periodic (a subclass of a.p.), odd, and possesses specific integral properties. This function is used to define non-self-adjoint and self-adjoint operators of the form $u'' \pm 2b(x)u'$.
4. Transformation to Self-Adjoint Form: To analyze the spectral properties, the non-self-adjoint operators are transformed into self-adjoint Schrödinger-type operators via an exponential change of variables ($v = e^{\int b} u$).
5. Limit Operator Analysis: The paper constructs sequences of translations of the coefficients to define limit operators. It analyzes the asymptotic behavior of the constructed functions to determine the criticality of these limit operators.

Key Contributions and Results
The paper presents three primary results that clarify the distinctions between the periodic and almost periodic settings:

1. Liouville-type Result for A.P. Operators:
   The paper establishes that for a self-adjoint operator with regular coefficients admitting a positive a.p. solution, the operator is critical if and only if the spatial dimension $N \le 2$ (Corollary 2.4). This extends known results for periodic operators to the almost periodic case under the assumption of an a.p. solution.

2. Non-Equivalence of Criticality and A.P. Principal Eigenvalue (Counter-example 1):
   The author constructs a self-adjoint a.p. operator in dimension $N=1$ that is critical but does not admit any positive a.p. eigenfunction.

   • Mechanism: The operator $L = \partial_x^2 - 2b(x)\partial_x$ admits a bounded positive solution $\phi(x) = e^{-\int_0^x b(t)dt}$ which decays to zero at infinity.
   • Result: Since $\phi$ is the unique positive solution (up to scalar multiples) and it is not a.p. (due to decay), the operator is critical but lacks an a.p. principal eigenvalue. This demonstrates that criticality does not imply the existence of an a.p. principal eigenvalue, even in dimension 1.

3. Instability of Criticality under Limits (Counter-example 2):
   The paper exhibits a self-adjoint a.p. operator $L$ in dimension $N=1$ that is subcritical with $\lambda_1 = 0$, yet admits a limit operator $L^*$ that is critical.

   • Mechanism: By analyzing the limit of translated coefficients of the function $b(x)$, a limit operator $L^*$ is constructed which admits a positive bounded solution.
   • Result: This shows that criticality is not an invariant property under the passage to limit operators in the almost periodic setting. A subcritical operator can have a critical limit, and conversely (as noted in Remark 1), a critical operator can have a subcritical limit.

Significance and Claims
The paper claims to reveal the "instability character of the criticality property in the almost periodic setting." Unlike the periodic case, where the criticality character is shared by all limit operators (which are merely translations), the almost periodic setting allows for a divergence between an operator and its limits.

Specifically, the paper highlights three striking differences between periodic and almost periodic operators:

1. An a.p. operator with $\lambda_1 = 0$ may not be critical (even in dimension 1), whereas for periodic operators, $L + \lambda_1$ is always critical in $N \le 2$.
2. The existence of an a.p. principal eigenvalue is not guaranteed for critical a.p. operators.
3. Criticality is not preserved under the formation of limit operators.

The work serves as a review and extension of the generalised principal eigenvalue theory, clarifying that the structural properties of periodic operators (such as the equivalence of criticality and the existence of a periodic eigenfunction in low dimensions) do not automatically extend to the almost periodic case. The results rely on elementary constructions and properties of limit periodic functions rather than probabilistic tools, offering a PDE-centric perspective on these spectral phenomena.