Absorbing Boundary Conditions for the Schrödinger Equation

Abstract

A large number of differential equation problems which admit traveling waves are usually defined on very large or infinite domains. To numerically solve these problems on smaller subdomains of the original domain, artificial boundary conditions must be defined for these subdomains. One type of artificial boundary condition which can minimize the size of such subdomains is the absorbing boundary condition. The imposition of absorbing boundary conditions is a technique used to reduce the necessary spatial domain when numerically solving partial differential equations that admit traveling waves. Such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations. In this paper, general absorbing boundary conditions will be developed for the Schrödinger equation with one spatial dimension, using group velocity considerations. Previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition. The well-posedness of the initial boundary value problem of the absorbing boundary condition, coupled to the interior Schrödinger equation, will also be discussed. Extension of the general absorbing boundary condition to higher spatial dimensions will be demonstrated. Numerical simulations using initial single Gaussian, double Gaussian, and a narrow Gaussian pulse distributions will be given, with comparision to exact solutions, to demonstrate the reflectivity properties of various orders of the absorbing boundary condition.

Sparse grids for the Schrödinger equation

“…Using the definition of the index setΩ K,T in (23) and evaluating the maximum in (27) we obtain the desired result (25).…”

“…We now plug this into (25) [38]. This behavior of the constant has of course to be compared with the behavior of ψ H 2,0 mix which also depends on N .…”

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

“…These ABC have been studied in [1], [2], [3], [6], [8]. The main idea is to absorb the components of the solution traveling with certain velocities.…”

“…That explains why some numerical experiments show a bad behavior when the initial value does not vanish at the boundary of the computational domain (see [8]) or when the ABC is not suitable to absorb the solution of (1.1).…”

Spectral Collocation Technique for Absorbing Boundary Conditions with Increasingly High Order Approximation

“…To attain a method with low computational cost, another way is to construct ABCs [9,10,11,12,13] by approximating the nonlocal operator in TBC with polynomials. This class of boundary conditions is local in time, and easy to implement.…”