Self-adjoint extensions of quantum graphs with eigenparameter-dependent vertex conditions: An operator-theoretic and symplectic geometry approach

We studied differential operators on compact metric graphs subject to vertex conditions that depend linearly on the spectral parameter. Such eigenparameter-dependent conditions arise naturally in several models of mathematical physics but complicate the standard operator-theoretic formulation of the problem. To address this difficulty, we introduced an extended Hilbert space framework that converts the original boundary value problem into an equivalent eigenvalue problem for an operator acting on a larger space. Within this setting, we established a characterization of self-adjoint realizations in terms of algebraic conditions on the matrices defining the vertex relations. The analysis was further interpreted using Hermitian symplectic geometry, which provides a natural description of admissible boundary conditions. As a consequence of the self-adjointness results, we showed that the associated operator possesses a compact resolvent and therefore has a purely discrete spectrum consisting of real eigenvalues with finite multiplicities. An illustrative example on a star graph was included to demonstrate the applicability of the framework and the resulting spectral properties. We also established connections to time-dependent problems: through Fourier analysis, eigenparameter-dependent conditions become first-order dynamic conditions for the heat equation, second-order conditions for the wave equation, and quantum dot models for the Schrödinger equation. Our framework unifies operator-theoretic and symplectic geometric approaches, providing verifiable criteria for self-adjointness applicable to arbitrary compact metric graphs. The results extend existing approaches for interval and simple graph models to general compact metric graphs and provide a systematic method for treating eigenparameter-dependent vertex conditions in the spectral theory of quantum graphs.