Quantum Graph Spectrum

Quantum graphs: A metric graph where each edge carries a 1D Schrödinger equation -ψ'' = k²ψ. At each vertex v of degree d_v, matching conditions enforce: (1) continuity: ψ_e(v) = ψ_e'(v) for all edges, (2) current conservation: ∑_e ψ'_e(v) = 0 (Neumann/Kirchhoff).

Secular equation: The spectrum is found by requiring the secular determinant det M(k) = 0, where M encodes the matching conditions. For a graph with E edges and L total length, the density of states oscillates around L/π with corrections from closed orbits (trace formula).

Level statistics: Quantum graphs show a crossover in level statistics from Poisson (integrable = straight wires) to GOE Wigner-Dyson (chaotic = complex graph). The Wigner surmise P(s) = (π/2)s·exp(-πs²/4) shows level repulsion; Poisson P(s) = exp(-s) has no repulsion.

Trace formula: Like the Gutzwiller formula for billiards, quantum graph spectra encode closed orbits (periodic paths on the graph) via ρ(k) ∝ ∑_{periodic orbits} A_p cos(k·L_p).