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Absorbing boundary conditions for the two-dimensional Schrodinger equation with an exterior potential.
 

Summary: Absorbing boundary conditions for the two-dimensional
Schr¨odinger equation with an exterior potential.
Part I: construction and a priori estimates
Xavier Antoine
Christophe Besse
Pauline Klein§
Abstract
The aim of this paper is to construct some classes of absorbing boundary conditions for the
two-dimensional Schr¨odinger equation with a time and space varying exterior potential and for
general convex smooth boundaries. The construction is based on asymptotics of the inhomogeneous
pseudodifferential operators defining the related Dirichlet-to-Neumann operator. Furthermore, a
priori estimates are developed for the truncated problems with various increasing order boundary
conditions. The effective numerical approximation will be treated in a second paper.
Contents
1 Introduction 2
2 What we already know and what remains true compared to the one-dimensional
case 4
2.1 The half-space case and a null potential . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The time dependent potential case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Specific aspects of the two-dimensional case 5

  

Source: Antoine, Xavier - Institut de Mathématiques Élie Cartan, Université Henri Poincaré - Nancy 1

 

Collections: Mathematics