Nature of the Singularities in the Spectrum of a One-Dimensional Ionic Lattice

It is known, by a theorem of van Hove, that regular lattices will have singularities in their frequency spectra, the type of singularity depending on the dimension. Although several studies of the one-dimensional lattice with Coulomb interactions have appeared, no one yet has succeeded in identifying the type of singularity that appears in addition to the inverse square root singularity, in the frequency spectrum. In this note we establish that for a one-dimensional crystal the singularity is of the form $g\left(\omega \right)\sim A{\left\{\right(\omega -{\omega }_1)\times \ln [\frac{1}{(\omega -{\omega }_1)}\left]\right\}}^{-\frac{1}{2}}$ where $A$ is a constant and ${\omega }_1$ is the position of the singularity.