High Order Perturbation Theory for Helmholtz/Schrödinger Equations via a Separable Preconditioner
✍ Scribed by Åke Edlund; Ilya Vorobeichik; Uri Peskin
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 345 KB
- Volume
- 138
- Category
- Article
- ISSN
- 0021-9991
No coin nor oath required. For personal study only.
✦ Synopsis
A numerical procedure is suggested for the solution of multidimensional inhomogeneous Helmholtz/Schro ¨dinger equations. The procedure is based on coordinate-space (grid) representations in which all the coupling terms (V ˆ) between different degrees of freedom are local (diagonal) and therefore the remaining differential (nonlocal) terms are separable. This separability leads to an efficient (sparse) representation of an approximate Green's operator (G ˆ0 ). For sufficiently ''weak'' coupling, a low order expansion in powers of G ˆ0 V ˆprovides the solution according to Rayleigh Schro ¨dinger perturbation theory. For ''strong'' coupling intensities the sparse structure of G ˆ0 makes it an efficient preconditioner for high order iterative solutions (e.g., the QMR algorithm of Freund and Nachtigal). The high order power expansion in G ˆ0 V provides an optimized perturbative series which converges for strong coupling intensities. A numerical example is given for the Helmholtz equation in three dimensions.