Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions

Virginie Bonnaillie-Noël (1) , Monique Dauge (1) , Nicolas Popoff (1) , Nicolas Raymond (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle \theta with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit \theta \to 0 and we prove full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.

Mots clés: Agmon estimates; Born-Oppenheimer approximation; Schrödinger operator; Semi classical limit

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff, Nicolas Raymond. Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions. Zeitschrift für Angewandte Mathematik und Physik, 2012, 63 (2), pp.203-231. ⟨10.1007/s00033-011-0163-y⟩. ⟨hal-00527643v4⟩