Exact energy bands and Fermi surfaces of separable Abelian potentials

We present a general theory for multidimensional Schrödinger equations with separable Abelian potentials with an arbitrary number of gaps in the spectrum. In particular, we derive general equations which allow one to express the energy and the wavevectors in the Brillouin zone as a function of the spectral parameters. By using the solutions of these equations, we show how to construct the energy bands and the Fermi surfaces in the first Brillouin zone of the reciprocal lattice. As illustrative examples we consider the case of two-dimensional separable potentials with one, two arid three gaps in the spectrum. The method can be applied to crystals with a cubic or a rectangular parallelogram Wigner-Seitz cell in arbitrary dimensions. The possibility to generalize the theory to other crystal symmetries is also briefly discussed.