= Solution
For $\alpha>0$, the dispersion is periodic and has minima at $k=0,\pi$ in the rotated full-zone convention, with
$$
\boxed{\epsilon_{\min}=2JS\sqrt{(1+\alpha)^2-1}=2JS\sqrt{\alpha(2+\alpha)}.}
$$
The maximum is $2JS(1+\alpha)$ at $k=\pm\pi/2$. The easy-axis state has no continuously broken <symmetry>, so a gap is allowed. Near a minimum the energy behaves as $2JS\sqrt{\alpha(2+\alpha)+(\delta k)^2}$. At $\alpha=0$ it becomes \b[$\epsilon_k=2JS|\sin k|$], linear near its zeros, with small-$k$ slope $2JS$ in the chosen units. Restoring lattice spacing $a$ and $\hbar$ gives spin-wave speed $2JSa/\hbar$. Zone folding reorganizes the same modes into the magnetic <Brillouin zone>.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2016/paper-1-spin-wave-spectra.png]
{title=Consistent easy-axis, isotropic and easy-plane linear spin-wave spectra; illustrative anisotropies}
These are linear spin-wave predictions, not a statement that the exact one-dimensional quantum chain always has long-range Néel order. At the isotropic point the zero-point <spin> reduction contains $N^{-1}\sum_k(|\sin k|^{-1}-1)/2$, which diverges logarithmically as the infrared cutoff vanishes. This failure of the assumed ordered reference must not be mistaken for a calculation of the exact interacting spectrum.
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