Density of states of a monatomic harmonic chain ($g(\omega )=2N/[\pi \sqrt{{\omega }_m^2-{\omega }^2}]$)

For the nearest-neighbour monatomic-chain dispersion relation, ${\omega }_m=2\sqrt{\alpha /M}$ and $d\omega /dq=(a/2)\sqrt{{\omega }_m^2-{\omega }^2}$ on the positive half-zone. The folded wave-number count gives the displayed density of states, including both propagation directions. Its upper-edge Van Hove singularity is integrable: ${\int }_0^{{\omega }_m}g(\omega )d\omega =N$. A finite chain has discrete modes; the continuous density describes its large-size limit.