Theory of glass dynamics: the low-lying modes of anharmonic materials

We study the lowest-lying excited states of arbitrary anharmonic solids (crystalline, glassy, or amorphous). Defining the harmonic ("skeleton") lattice underlying any such anharmonic medium, we find that its spectrum of elementary excitations provides a rigorous upper bound to that of the anharmonic medium. We then calculate upper bounds to that spectrum, in terms of an isotropic, homogeneous, harmonic medium.