Bound states in a nonlinear Kronig - Penney model

We study the bound states of a Kronig Penney potential for a nonlinear one-dimensional Schrödinger equation. This potential consists of a large, but not necessarily infinite, number of equidistant -function wells. We show that the ground state can be highly degenerate. Under certain conditions furthermore, even the bound state that would normally be the highest can have almost the same energy as the ground state. This holds for other simple periodic potentials as well.