Zeitschrift für angewandte Mathematik und Physik

, Volume 63, Issue 2, pp 203–231

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

Authors

    • IRMAR, ENS Cachan BretagneUniversité de Rennes 1, CNRS, UEB
  • Monique Dauge
    • IRMARUniversité de Rennes 1, CNRS
  • Nicolas Popoff
    • IRMARUniversité de Rennes 1, CNRS
  • Nicolas Raymond
    • IRMARUniversité de Rennes 1, CNRS
Article

DOI: 10.1007/s00033-011-0163-y

Cite this article as:
Bonnaillie-Noël, V., Dauge, M., Popoff, N. et al. Z. Angew. Math. Phys. (2012) 63: 203. doi:10.1007/s00033-011-0163-y

Abstract

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 that makes the angle θ 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 θ → 0, and we prove a 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.

Mathematics Subject Classification (2000)

Primary 99Z99Secondary 00A0035P1535P2041A6065N2565N30

Keywords

Agmon estimatesBorn-Oppenheimer approximationSchrödinger operatorSemi classical limit

Copyright information

© Springer Basel AG 2011