= Solution
Write $h=\Delta x$ and set $u_0=u_{M+1}=0$. In the discrete <L2 norm> $\|u\|_h^2=h\sum_{m=1}^M|u_m|^2$, the <energy method> gives
$$
\frac12\frac{d}{dt}\|u\|_h^2
=-\frac1h\sum_{m=0}^M|u_{m+1}-u_m|^2\leq0.
$$
To obtain this identity, shift the indices in the centered second-difference sum. The centered first-difference <matrix> is <skew-symmetric>, so its contribution has zero real part even for complex grid values. The homogeneous <Dirichlet boundary conditions> remove the endpoint terms.
Therefore the semidiscretization is \b[stable for every real $\alpha$ and every positive $h$], with the mesh-independent bound $\|u(t)\|_h\leq\|u(0)\|_h$. No restriction on $|\alpha|h$ is needed for this energy estimate.
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