As I remember from a quantum mechanics course I took in the 1940s, there is a way of estimating energy levels that does not solve the wave equation. I learned it only for discrete energy levels, but I'd suppose it applies to bands. It is based on the fact that the energy level is a minimum of an energy integral over all functions satisfying certain conditions. The method consists of representing a hypothetical wave function by a linear combination of terms satisfying boundary conditions. You then choose coefficients to minimize the energy which is a sum of products. As I recall it was used to estimate the energy levels of the helium atom and the ionized hydrogen molecule. You get an upper bound, and sometimes it was quite close to the experimental value.