Title: Schrödinger operators periodic in octants,
Abstract: We consider Schrödinger operators with periodic potentials in the positive
quadrant for dim $>1$ with Dirichlet boundary condition. We show that for any
integer $N$ and any interval $I$ there exists a periodic potential such that
the Schrödinger operator has $N$ eigenvalues counted with the multiplicity on
this interval and there is no other spectrum on the interval. Furthermore, to
the right and to the left of it there is a essential spectrum.
Moreover, we prove similar results for Schrödinger operators for other
domains. The proof is based on the inverse spectral theory for Hill operators
on the real line.