We provide a complete Sturm--Liouville spectral analysis of the Constant Elasticity of Variance (CEV) operator. By transforming the corresponding Fokker--Planck operator into a generalized Laguerre operator, we explicitly characterize its self-adjoint extensions, boundary conditions, spectra, and eigenfunctions across all elasticity regimes. We then relate these spectral features to arbitrage phenomena in the CEV model, showing how boundary behavior and positive harmonic functions encode the distinction between attainable-boundary arbitrage mechanisms and strict-local-martingale bubble regimes. The result is an explicit operator-theoretic perspective on the link between CEV dynamics, no-arbitrage, and spectral theory.