Quantum Graph: Star Topology
Secular equation · eigenvalue spectrum · wave functions on edges
Quantum graphs replace Schrödinger's equation on metric graphs — each edge of length Lⱼ carries a wave function ψⱼ(x) = Aⱼ sin(kx) + Bⱼ cos(kx). At vertices, Neumann-Kirchhoff conditions enforce continuity and current conservation: Σⱼ ψ'ⱼ(0) = 0 at the hub.
Secular equation (star graph): Σⱼ cot(k·Lⱼ) = 0
Eigenvalues kₙ are roots of this equation. When all Lⱼ are equal (commensurate), eigenvalues cluster; incommensurate lengths cause repulsion. The spectral statistics interpolate between Poisson (integrable) and GUE/GOE (chaotic) depending on the rationality of length ratios.