On the guided states of 3D biperiodic Schrödinger operators

François Bentosela (1) , Claude Bourrely (1) , Yves Dermenjian (2) , Eric Soccorsi (1)

1 CPT - Centre de Physique Théorique - UMR 7332
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We consider the Laplacian operator H_0 perturbed by a non-positive potential $V$, which is periodic in two directions, and decays in the remaining 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_0+V in the periodic variables. If V is sufficiently small and decreases fast enough in the infinite 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 faster than any polynomial function in the infinite direction.

Mots clés: Schrödinger operator; Periodic potential; Guided state; Limiting absorption principle

Domaines: Equations aux dérivées partielles [math.AP]

François Bentosela, Claude Bourrely, Yves Dermenjian, Eric Soccorsi. On the guided states of 3D biperiodic Schrödinger operators. Communications in Partial Differential Equations, 2012, 37 (10), pp.1805-1838. ⟨hal-00737389⟩