 Solution

The centred second difference has Fourier symbolWriting $q={\sin }^2(\theta /2)$ givesFor the denominator to stay positive and $|H|\le 1$, one needsThe right inequality follows from $\mu \ge 0$, while the left is equivalent to $2-4\alpha q\ge 0$. Hence stability for every Fourier mode isapart 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.