A model of a variable-band semiconductor which permits of an accurate solution is considered. The wave functions and an equation defining the energy spectrum of the system are obtained. The local density of states, which, generally speaking, is an oscillating function depending on the energy and number of lattice points is calculated. An accurate representation is introduced of the coordinate dependence of the region of the maximum of the local density of states, which is a generalization of the approximate representation of the curvature of the energy bands. In the case of a fairly smooth variation in the potential parameter of the variable-band system a comparison is made with the approximate band-structure pattern.