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Let $H_h$ be the periodic centered second-difference matrix minus the real diagonal matrix containing $V(x_m)$. It is a Hermitian matrix, so the semidiscrete system iswith a skew-Hermitian matrix generator. ConsequentlyIts exact propagator $e^{-it{H}_h}$ is a unitary matrix, and therefore the semidiscretization is stable in the discrete $2$-norm, uniformly for all times and mesh sizes.

Write the semidiscrete system as ${\mathbf{u}}^{\prime }=L_h\mathbf{u}+\alpha D_h\mathbf{u}$. With zero boundary values, the centered second-difference matrix $L_h$ is symmetric negative definite and the centered first-difference matrix $D_h$ is skew-symmetric. ThereforeThis is a mesh-uniform stability estimate for the centered convection-diffusion semidiscretization, valid for every real $\alpha$.

Both centered differences have local spatial error $O(h^2)$ for a sufficiently smooth solution. Stability plus consistency gives convergence, equivalently by the semidiscrete form of the Lax equivalence theorem. Thus