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Repeat the linear spin-wave approximation for bonds of lengths one and two. The polarized energy is $E_0=-N{S}^2(J_1+J_2)$ and each bond length contributes its own lattice Laplacian. ThusThe next-nearest-neighbour ferromagnetic magnon dispersion remains nonnegative for the stated positive exchanges and has its only zero at $k=0$ modulo $2\pi$ when $J_1>0$. At small $k$, ${\epsilon }_k\simeq S(J_1+4J_2)k^2$: longer-range ferromagnetic exchange increases the stiffness.

At this harmonic order, a bosonic occupation configuration has energy $E_0+{\sum }_k{\epsilon }_k{n}_k$. The one-magnon dispersion is also exact. General multi-magnon energies acquire the interactions omitted in the large-$S$ expansion; the formula does not claim an exact diagonalization of the full interacting many-spin Hamiltonian.