Schrödinger operators periodic in octants

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