Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

Anton Arnold

doi:10.1155/1998/38298

Abstract

Transparent boundary conditions for the transient Schrödinger equation on a domain Ω can be derived explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these boundary conditions are non‐local in time (of memory type). For the Crank‐Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary conditions over existing ad‐hoc discretizations of the differential transparent boundary conditions. As an application of these boundary conditions to the modeling of quantum devices, a transient 1D scattering model for mixed quantum states is presented.

Highlights

The formulation and implementation of physically reasonable and mathematically well-posed boundary conditions (BC) is one of the big open problems and challenges for transient simulations of semiconductor devices through quantum mechanical models
If the initial data is supported on this finite domain D, one can approximate lfa, the exact solution of the whole space problem restricted to D, by solving the original problem only on D, together with ABC’s on /)if2
We conjecture that the resulting initial boundary value problems (IBVP) is well-posed; this problem will be analyzed in a forthcoming paper

Summary

INTRODUCTION

The formulation and implementation of physically reasonable and mathematically well-posed boundary conditions (BC) is one of the big open problems and challenges for transient simulations of semiconductor devices through quantum mechanical models. If the initial data is supported on this finite domain D,, one can approximate lfa, the exact solution of the whole space problem restricted to D,, by solving the original problem only on D,, together with ABC’s on /)if2 If this approximate solution coincides on f with the exact solution, one refers to these BC’s as transparent boundary conditions (TBC). Several heuristic BC’s were introduced for the SE, and they yield reasonable results for either short time calculations or a limited frequency range These existing strategies include the introduction of an artificial absorption or attenuation layer close to the boundary, which can be interpreted as adding a complex potential in (1.1) (see 11 ], 19], [20]).

TRANSPARENT BOUNDARY CONDITIONS

DISCRETIZATION OF THE TRANSPARENT BOUNDARY CONDITION IN 1D

A TRANSIENT SCHRODINGER-POISSON SCATTERING MODEL WITH TBC’S