Spectral Phase Transitions in Rank-One Nonlocal Elliptic Operators: Global Coupling, Nonlinear Bifurcation, and Geometric Extensions

This work presents a complete spectral and dynamical analysis of elliptic operators with a rank-one global coupling. Such operators arise naturally in constrained variational problems, where a single global degree of freedom induces nonlocal effects while leaving local force-based dynamics unchanged. We derive the exact spectral structure of the operator, showing that the rank-one coupling affects only a single global mode while preserving self-adjointness and compactness of the resolvent. A codimension-one spectral phase transition is identified, characterized by the emergence of a zero eigenvalue and a controlled singularity of the resolvent at a critical coupling strength. The analysis is extended to nonlinear settings via Lyapunov–Schmidt reduction, revealing that the global mode governs the bifurcation structure while higher modes remain slaved. A dynamical extension using center manifold theory establishes the effective low-dimensional dynamics near the critical regime. Finally, the framework is generalized to compact Riemannian manifolds, highlighting the role of geometry and topology in the global spectral behavior. The results provide a rigorous and self-contained classification of operators with a single global degree of freedom and are of independent interest in spectral theory, nonlocal elliptic equations, and constrained variational analysis.