The methods of Bethe and Hulth\'en are used to build spin-wave states for the antiferromagnetic linear chain. These states, of spin 1 and translational quantum number
k, are eigenstates of the Hamiltonian $H=\ensuremath{\Sigma}{j}^{}{\mathrm{S}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\ensuremath{\epsilon}}_{k}=(\frac{\ensuremath{\pi}}{2})|sink|$, whereas Anderson's spin-wave theory gives ${\ensuremath{\epsilon}}_{k}=|sink|$. For finite chains it has been verified by numerical computation that these states are the lowest states of given
k, but no rigorous proof has been given for an infinite chain.