= Density of states of a monatomic harmonic chain
{title2=$g(\omega)=2N/[\pi\sqrt{\omega_m^2-\omega^2}]$}
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.
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