Global Recursion Polynomial Expansions of the Green’s Function and Time Evolution Operator for the Schrödinger Equation with Absorbing Boundary Conditions

A detailed analysis of absorbing boundary conditions for the linear Schrodinger equation is presented in this paper. It is focused on absorbing boundary conditions that are obtained by using rational functions to approximate the exact transparent boundary conditions. Different strategies are investigated for the optimal selection of the coefficients of these rational functions, including the Pade approximation, the L2 norm approximations of the Fourier symbol, L2 minimization of a reflection coefficient, and two error minimization techniques for the chosen benchmark problems with known exact solutions. The results of computational experiments are given and a detailed comparison of the efficiency of these techniques is presented.

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Absorbing boundary conditions on a hypersurface for the Schrödinger equation in a half-space

Applied Mathematics Letters ◽

10.1016/0893-9659(96)00051-1 ◽

1996 ◽

Vol 9 (4) ◽

pp. 55-59 ◽

Cited By ~ 6

Author(s):

L. Di Menza

Keyword(s):

Schrödinger Equation ◽

Boundary Conditions ◽

Half Space ◽

Schrodinger Equation ◽

Absorbing Boundary Conditions ◽

Absorbing Boundary

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SOLVING THE TIME-DEPENDENT SCHRÖDINGER EQUATION WITH ABSORBING BOUNDARY CONDITIONS AND SOURCE TERMS IN MATHEMATICA 6.0

International Journal of Modern Physics C ◽

10.1142/s0129183110015919 ◽

2010 ◽

Vol 21 (11) ◽

pp. 1391-1406 ◽

Cited By ~ 5

Author(s):

F. L. DUBEIBE

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Evolution Operator ◽

Matrix Inversion ◽

Absorbing Boundary Conditions ◽

Time Dependent ◽

Source Terms ◽

Absorbing Boundary ◽

Simple Method ◽

Numerical Computing

In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrödinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into these schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrödinger equation by using a standard Crank–Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N × N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.

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ABSORBING BOUNDARY CONDITIONS FOR THE TWO-DIMENSIONAL SCHRÖDINGER EQUATION WITH AN EXTERIOR POTENTIAL PART I: CONSTRUCTION AND A PRIORI ESTIMATES

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202512500261 ◽

2012 ◽

Vol 22 (10) ◽

pp. 1250026 ◽

Cited By ~ 17

Author(s):

XAVIER ANTOINE ◽

CHRISTOPHE BESSE ◽

PAULINE KLEIN

Keyword(s):

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

A Priori Estimates ◽

Pseudodifferential Operators ◽

A Priori ◽

Absorbing Boundary Conditions ◽

Two Dimensional ◽

Absorbing Boundary ◽

Priori Estimates

The aim of this paper is to construct some classes of absorbing boundary conditions for the two-dimensional Schrödinger 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.

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