Spectral Analysis of the Möbius Laplacian

Abstract:We study the Laplacian on the Möbius strip, realized as the quotient of the flat cylinder S¹ × [−1, 1] by the free Z/2 action ψ(s, t) = (s + π, −t). The operator itself is the ordinary flat Laplacian Δu = u_ss + u_tt; non-orientability enters only through the domain, i.e. through the requirement u∘ψ = u. Fixing Dirichlet boundary conditions on the free edges t = ±1, we derive the explicit spectrum λ_k,n = k² + (nπ/2)², restricted to the pairs (k even, n odd) or (k odd, n even), k ∈ Z, n ≥ 1. We confirm this formula by two independent methods — closed-form separation of variables, and a full two-dimensional finite-difference discretization with an explicit symmetry projection — which agree to within expected discretization error. The ground state is λ₁ = π²/4, numerically identical to that of the ordinary cylinder under the same boundary condition: the spectral gap is inherited from the Dirichlet condition, not created by the twist. The twist's genuine, verified effect is to remove roughly half of the cylinder's higher modes — confirmed explicitly on a forbidden test case (k = 0, n = 2). We do not claim, and this note does not establish, any connection to the Yang–Mills mass gap, Navier–Stokes regularity, or the Riemann Hypothesis.