Schrödinger operators periodic in octants

We consider Schrodinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrodinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrodinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.