  The dispersion relations of energy bands in solids are characterized by their
density of states, but a given density of states may originate from various
band structures. We show how a spherically symmetric dispersion can be
constructed for any one-band density of states. This method is applied to one-,
two- and three-dimensional systems. It also serves to establish that any
one-band spectrum with finite bandwidth can be obtained from a properly scaled
dispersion relation in the limit of infinite dimensions.
