Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh--Weyl theory
- Ehrhardt, Matthias
- Zheng, Chunxiong
2010 Mathematics Subject Classification
- 65M99 81-08
- absorbing boundary conditions, variable potential, Schrödinger equation, Titchmarsh-Weyl m-function, unbounded domain
A new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band.