On the Guided States of 3D Biperiodic Schrödinger Operators

We consider the Laplacian operator H ₀: = ? Δ perturbed by a non-positive potential V, which is periodic in two directions, and decays in the third one. We are interested in the characterization and decay properties of the guided states, defined as the eigenfunctions of the reduced operators in the Bloch-Floquet-Gelfand transform of H: = H ₀ + V in the periodic variables. If V is sufficiently small and decreases fast enough in the third direction, we prove that, generically, these guided states are characterized by quasi-momenta belonging to some one-dimensional compact real analytic submanifold of the Brillouin zone. Moreover they decay in the third direction faster than any rational function without real pole.