21773 articles – 15587 references  [version française]

hal-00527643, version 3

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

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 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 a two term asymptotics 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.

  • 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
  • CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
  • Domain : Mathematics/Analysis of PDEs
    Physics/Mathematical Physics
    Mathematics/Mathematical Physics
  • Keywords : Agmon estimates – Born-Oppenheimer approximation – Schrödinger operator – Semi classical limit
  • Internal note : 2010-64
  • Available versions :  v1 (2010-10-20) v2 (2010-10-20) v3 (2010-11-22) v4 (2011-06-18)
 
  • hal-00527643, version 3
  • http://hal.archives-ouvertes.fr/hal-00527643
  • oai:hal.archives-ouvertes.fr:hal-00527643
  • From: Monique Dauge <>
  • Submitted on: Monday, 22 November 2010 10:07:34
  • Updated on: Thursday, 13 January 2011 17:53:48