发表机构
Department of Mathematics, International University, Ho Chi Minh City, Vietnam; Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam（国际大学数学系; 越南国立大学胡志明市）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对带固定符号漂移的一维非局部扩散算子，通过直接方法和多种分析技术确定其主特征值的精确标度极限与谱相变，填补了非自伴情形下变分刻画缺失的研究空白。
AI 中文摘要
我们研究具有固定符号漂移的一维非局部扩散算子的广义主特征值。该非自伴情形的一个主要困难是主值缺乏变分刻画。在对称无漂移问题中，临界非局部到局部极限可通过二次变分结构和Sobolev半范数逼近处理，如Berestycki-Coville-Vo的研究。漂移破坏了该结构，且在有界区间上同时引入了单侧流入-流出边界几何。因此，用在奇异重标度下保持稳定的估计替代缺失的变分论证是核心技术问题。我们的方法是直接的：处理单个广义主特征值，从方程本身证明最大值原理、正锥中的简单性以及标度极限，不经过类似Berestycki-Rossi研究中的辅助广义主特征值λ₁'和λ₁''。替代机制包括方向Harnack不等式、系数障碍、直接-伴随恒等式、对数Collatz-Wielandt变换、零延拓的Fourier强制性以及依赖标度的局域化。在临界扩散标度下，Fourier紧性和精确能量-输运恒等式恢复了缺失的流入Dirichlet条件，无需Rayleigh商即可确定局部Dirichlet极限。对于变系数，我们确定了有界区间和整个直线上的完整小范围相图，并通过秩一约化和平流旅行时变量中的角Laplace分析得到了精确的大范围三项渐近式。在齐次全直线问题中，临界修正项的阶为σ²。
英文摘要
We consider generalized principal eigenvalues of one-dimensional nonlocal dispersal operators with a drift of fixed sign. A principal difficulty in this non-self-adjoint setting is the absence of a variational characterization of the principal value. In the symmetric drift-free problem, the critical nonlocal-to-local limit can be treated through a quadratic variational structure and Sobolev-seminorm approximation as Berestycki-Coville-Vo \cite{BCV}. The drift destroys this structure and, on a bounded interval, introduces at the same time a one-sided inflow--outflow boundary geometry. Replacing the missing variational argument by estimates which remain stable under singular rescaling is therefore a central technical issue. Our approach is direct : we work with a single generalized principal value and prove the maximum principle, simplicity in the positive cone and the scaling limits from the equation itself, without passing through auxiliary generalized principal eigenvalues analogous to $λ_1'$ and $λ_1''$ in Berestycki--Rossi \cite{BR}. The replacement mechanisms are directional Harnack inequalities and coefficient barriers, direct--adjoint identities, logarithmic Collatz--Wielandt transforms, Fourier coercivity of zero extensions and scale-dependent localization. At the critical diffusive scale, Fourier compactness and an exact energy--transport identity recover the missing inflow Dirichlet condition and identify the local Dirichlet limit without a Rayleigh quotient. For variable coefficients we determine the complete small-range phase diagram on bounded intervals and on the line, and we obtain sharp large-range three-term asymptotics by a rank-one reduction and a corner Laplace analysis in the advective travel-time variable. In the homogeneous whole-line problem the critical correction is of order $σ^2$.
