Volumetric fast multipole method for modeling Schrödinger's equation

Zhiqin Zhao; Narayan Kovvali; Wenbin Lin; Chang Hoi Ahn; Luise Couchman; Lawrence Carin

doi:10.1016/j.jcp.2006.11.003

Volumetric fast multipole method for modeling Schrödinger's equation

Zhiqin Zhao, Narayan Kovvali, Wenbin Lin, Chang Hoi Ahn, Luise Couchman, Lawrence Carin

Research output: Contribution to journal › Article › peer-review

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Abstract

A volume integral equation method is presented for solving Schrödinger's equation for three-dimensional quantum structures. The method is applicable to problems with arbitrary geometry and potential distribution, with unknowns required only in the part of the computational domain for which the potential is different from the background. Two different Green's functions are investigated based on different choices of the background medium. It is demonstrated that one of these choices is particularly advantageous in that it significantly reduces the storage and computational complexity. Solving the volume integral equation directly involves O(N2) complexity. In this paper, the volume integral equation is solved efficiently via a multi-level fast multipole method (MLFMM) implementation, requiring O(N log N) memory and computational cost. We demonstrate the effectiveness of this method for rectangular and spherical quantum wells, and the quantum harmonic oscillator, and present preliminary results of interest for multi-atom quantum phenomena. © 2006 Elsevier Inc. All rights reserved.

Original language	English (US)
Pages (from-to)	941-955
Number of pages	15
Journal	Journal of Computational Physics
Volume	224
Issue number	2
DOIs	https://doi.org/10.1016/j.jcp.2006.11.003
State	Published - Jun 10 2007
Externally published	Yes

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title = "Volumetric fast multipole method for modeling Schr{\"o}dinger's equation",

abstract = "A volume integral equation method is presented for solving Schr{\"o}dinger's equation for three-dimensional quantum structures. The method is applicable to problems with arbitrary geometry and potential distribution, with unknowns required only in the part of the computational domain for which the potential is different from the background. Two different Green's functions are investigated based on different choices of the background medium. It is demonstrated that one of these choices is particularly advantageous in that it significantly reduces the storage and computational complexity. Solving the volume integral equation directly involves O(N2) complexity. In this paper, the volume integral equation is solved efficiently via a multi-level fast multipole method (MLFMM) implementation, requiring O(N log N) memory and computational cost. We demonstrate the effectiveness of this method for rectangular and spherical quantum wells, and the quantum harmonic oscillator, and present preliminary results of interest for multi-atom quantum phenomena. {\textcopyright} 2006 Elsevier Inc. All rights reserved.",

author = "Zhiqin Zhao and Narayan Kovvali and Wenbin Lin and Ahn, {Chang Hoi} and Luise Couchman and Lawrence Carin",

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doi = "10.1016/j.jcp.2006.11.003",

language = "English (US)",

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T1 - Volumetric fast multipole method for modeling Schrödinger's equation

AU - Zhao, Zhiqin

AU - Kovvali, Narayan

AU - Lin, Wenbin

AU - Ahn, Chang Hoi

AU - Couchman, Luise

AU - Carin, Lawrence

N1 - Generated from Scopus record by KAUST IRTS on 2021-02-09

PY - 2007/6/10

Y1 - 2007/6/10

N2 - A volume integral equation method is presented for solving Schrödinger's equation for three-dimensional quantum structures. The method is applicable to problems with arbitrary geometry and potential distribution, with unknowns required only in the part of the computational domain for which the potential is different from the background. Two different Green's functions are investigated based on different choices of the background medium. It is demonstrated that one of these choices is particularly advantageous in that it significantly reduces the storage and computational complexity. Solving the volume integral equation directly involves O(N2) complexity. In this paper, the volume integral equation is solved efficiently via a multi-level fast multipole method (MLFMM) implementation, requiring O(N log N) memory and computational cost. We demonstrate the effectiveness of this method for rectangular and spherical quantum wells, and the quantum harmonic oscillator, and present preliminary results of interest for multi-atom quantum phenomena. © 2006 Elsevier Inc. All rights reserved.

AB - A volume integral equation method is presented for solving Schrödinger's equation for three-dimensional quantum structures. The method is applicable to problems with arbitrary geometry and potential distribution, with unknowns required only in the part of the computational domain for which the potential is different from the background. Two different Green's functions are investigated based on different choices of the background medium. It is demonstrated that one of these choices is particularly advantageous in that it significantly reduces the storage and computational complexity. Solving the volume integral equation directly involves O(N2) complexity. In this paper, the volume integral equation is solved efficiently via a multi-level fast multipole method (MLFMM) implementation, requiring O(N log N) memory and computational cost. We demonstrate the effectiveness of this method for rectangular and spherical quantum wells, and the quantum harmonic oscillator, and present preliminary results of interest for multi-atom quantum phenomena. © 2006 Elsevier Inc. All rights reserved.

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021999106005596

UR - http://www.scopus.com/inward/record.url?scp=34248591330&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2006.11.003

DO - 10.1016/j.jcp.2006.11.003

M3 - Article

SN - 1090-2716

VL - 224

SP - 941

EP - 955

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 2

ER -

Volumetric fast multipole method for modeling Schrödinger's equation

Abstract

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