Commun. Comput. Phys.,
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Volume 3.


An Exact Absorbing Boundary Condition for the Schrodinger Equation With Sinusoidal Potentials at Infinity

Chunxiong Zheng 1*

1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.

Received 9 April 2007; Accepted (in revised version) 2 June 2007
Available online 30 October 2007

Abstract

In this paper we study numerical issues related to the Schr\"odinger equation with sinusoidal potentials at infinity. An exact absorbing boundary condition in a form of Dirichlet-to-Neumann mapping is derived. This boundary condition is based on an analytical expression of the logarithmic derivative of the Floquet solution to Mathieu's equation, which is completely new to the author's knowledge. The implementation of this exact boundary condition is discussed, and a fast evaluation method is used to reduce the computation burden arising from the involved half-order derivative operator. Some numerical tests are given to show the performance of the proposed absorbing boundary conditions.

AMS subject classifications: 65M99, 81-08

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Key words: Absorbing boundary condition, sinusoidal potential, Schr\"odinger equation, unbounded domain.

*Corresponding author.
Email: czheng@math.tsinghua.edu.cn (C. Zheng)
 

The Global Science Journal