Semiclassical density of states from periodic orbits in a chaotic billiard
The Gutzwiller trace formula (1971) connects quantum energy levels to classical periodic orbits: d(E) = d̄(E) + (1/πℏ) Σ_γ A_γ cos(S_γ/ℏ − μ_γ π/2). Each periodic orbit γ contributes an oscillation with amplitude A_γ ∝ 1/√|det(M_γ−I)| (stability) and phase S_γ/ℏ (action). Below: a rectangular billiard (integrable) and stadium billiard (chaotic) with their oscillating density of states built from periodic orbit sums.