= Solution
The centred second difference has Fourier symbol
$$
e^{-i\theta}-2+e^{i\theta}
=-4\sin^2\frac\theta2.
$$
Writing $q=\sin^2(\theta/2)$ gives
$$
\boxed{
H(\theta)=
\frac{1-2(\mu+\alpha)q}
{1+2(\mu-\alpha)q}
}.
$$
For the denominator to stay positive and $|H|\leq1$, one needs
$$
-\bigl[1+2(\mu-\alpha)q\bigr]
\leq1-2(\mu+\alpha)q
\leq1+2(\mu-\alpha)q.
$$
The right inequality follows from $\mu\geq0$, while the left is equivalent to $2-4\alpha q\geq0$. Hence stability for every Fourier mode is
$$
\boxed{0\leq\alpha\leq\frac12,\qquad \mu\geq0},
$$
apart from the singular endpoint $(\mu,\alpha)=(0,1/2)$, where both sides annihilate the $\theta=\pi$ mode and the recurrence does not determine its next value.
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