Absorbing Boundary Conditions for the Schrödinger Equation

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 Schrodinger 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 Schrodinger 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.

[1]  Thomas Fevens,et al.  ABSORBING BOUNDARY CONDITIONS FOR THE SCHR ¨ ODINGER EQUATION , 1999 .

[2]  Wilbert Lick,et al.  Open boundary conditions for hyperbolic equations , 1987 .

[3]  Reiko Sakamoto,et al.  Hyperbolic boundary value problems , 1982 .

[4]  R. Renaut,et al.  Abstract of papers to appear in future issuesAbsorbing boundary conditions, difference operators, and stability , 1992 .

[5]  G. R. Hadley,et al.  Transparent boundary condition for beam propagation. , 1991, Optics letters.

[6]  Eitan Abraham,et al.  Two‐dimensional time‐dependent quantum‐mechanical scattering event , 1984 .

[7]  Robert L. Higdon,et al.  Numerical absorbing boundary conditions for the wave equation , 1987 .

[8]  H. Kreiss.  Initial boundary value problems for hyperbolic systems , 1970 .

[9]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[10]  E. Lindman.  “Free-space” boundary conditions for the time dependent wave equation , 1975 .

[11]  R. Kosloff,et al.  Absorbing boundaries for wave propagation problems , 1986 .

[12]  Dale R. Durran,et al.  Toward More Accurate Wave-Permeable Boundary Conditions , 1993 .

[13]  K. K. Mei,et al.  Superabsorption-a method to improve absorbing boundary conditions (electromagnetic waves) , 1992 .

[14]  Alan J. Thorpe,et al.  Radiation conditions for the lateral boundaries of limited‐area numerical models , 1981 .

[15]  S. Orszag,et al.  Approximation of radiation boundary conditions , 1981 .

[16]  Björn Engquist,et al.  Absorbing boundary conditions for wave-equation migration , 1980 .

[17]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[18]  Mark Hedley,et al.  Radiation Boundary Conditions in Numerical Modeling , 1988 .

[19]  B. M. Fulk.  MATH , 1992 .

[20]  I. Orlanski.  A Simple Boundary Condition for Unbounded Hyperbolic Flows , 1976 .

[21]  R. Higdon.  Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation , 1986 .

[22]  R. Higdon.  Radiation boundary conditions for dispersive waves , 1994 .

[23]  Judah L. Schwartz,et al.  Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena , 1967 .

[24]  L. Trefethen.  Group velocity in finite difference schemes , 1981 .

[25]  L. Trefethen.  Group velocity interpretation of the stability theory of Gustafsson, Kreiss, and Sundström , 1983 .

[26]  T. Porsching,et al.  Numerical Analysis of Partial Differential Equations , 1990 .

[27]  R. G. Keys.  Absorbing boundary conditions for acoustic media , 1985 .

[28]  J. Broeze,et al.  Variational principles and conservation laws in the derivation of radiation boundary conditions for wave equations , 1992 .

[29]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[30]  J. Broeze,et al.  Radiation boundary conditions for the two-dimensional wave equation from a variational principle , 1992 .

[31]  Robert A. Pearson,et al.  Consistent Boundary Conditions for Numerical Models of Systems that Admit Dispersive Waves , 1974 .

[32]  Kuska.  Absorbing boundary conditions for the Schrödinger equation on finite intervals. , 1992, Physical review. B, Condensed matter.

[33]  Yau Shu Wong,et al.  Absorbing boundary conditions for second-order hyperbolic equations , 1990 .

[34]  M. Baer,et al.  The time‐dependent Schrödinger equation: Application of absorbing boundary conditions , 1989 .

[35]  L. Trefethen,et al.  Well-Posedness of one-way wave equations and absorbing boundary conditions , 1986 .

[36]  Robert L. Higdon,et al.  Radiation boundary conditions for elastic wave propagation , 1990 .

[37]  R. Higdon,et al.  Initial-boundary value problems for linear hyperbolic systems , 1986 .

[38]  G. R. Hadley,et al.  Transparent boundary condition for the beam propagation method , 1992 .

[39]  Rosemary A. Renaut,et al.  Abstract of papers to appear in future issuesAbsorbing boundary conditions, difference operators, and stability , 1992 .

[40]  William H. Raymond,et al.  A radiation boundary condition for multi‐dimensional flows , 1984 .

[41]  Shibata.  Absorbing boundary conditions for the finite-difference time-domain calculation of the one-dimensional Schrödinger equation. , 1991, Physical review. B, Condensed matter.