= Solution
Use $a_n=N^{-1/2}\sum_ke^{ikn}a_k$ with $k=2\pi j/N$, choosing $N$ distinct values modulo $2\pi$ in the <Brillouin zone>. The lattice spacing is one; for spacing $a$ the dimensionless variable here would be $ka$. Orthogonality of the discrete Fourier modes diagonalizes the bilinear Hamiltonian:
$$
H=E_0+\sum_k\varepsilon_k a_k^\dagger a_k,\qquad
\boxed{\varepsilon_k=2JS(1-\cos k)=4JS\sin^2(k/2).}
$$
This is the <nearest-neighbour ferromagnetic magnon dispersion>. The printed expression is missing a factor of four for the stated exchange Hamiltonian. If $\omega_k$ denotes a physical frequency, $\hbar\omega_k=\varepsilon_k$; setting $\hbar=1$ does not remove that factor. The printed coefficient would require an exchange constant $J/4$ in the original Hamiltonian.
An independent check uses one spin lowering at site $n$. The exact one-magnon Hamiltonian relative to $E_0$ has diagonal element $2JS$ and off-diagonal elements $-JS$ at $n\pm1$, giving the same Fourier eigenvalue. At small $k$, \b[$\varepsilon_k\simeq JS k^2$], and the $k=0$ state belongs to the degenerate ground multiplet. The quadratic low-energy branch is the ferromagnetic <type-B Goldstone boson>.
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