2007/04/22 by Simen Kvaal, Morten Hjorth-Jensen, M. Hjorth‐Jensen +2
Engineering · Mathematics · Physics and Astronomy · #Electromagnetic Simulation and Numerical Methods #Numerical methods for differential equations #Quantum and electron transport phenomena #cond-mat.str-el #nucl-th #physics.comp-ph
paper · pdf · doi:10.1103/physrevb.76.085421
published as Phys. Rev. B 76 085421 (2007) · 7 figures, submitted to Physical Review B Single reference error fixed
arxiv created 2007/04/22 · openalex publication_date 2007/08/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method [S. Gould, Variational Methods for Eigenvalue Problems: An Introduction to the Methods of Rayleigh, Ritz, Weinstein, and Aronszajn (Dover, New York, 1995)], also called a spectral method, configuration-interaction method, or ``exact diagonalization'' method) is systematically analyzed using results for the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensional model of a quantum dot. The method is shown to converge slowly, and the nonsmooth character of the interaction potential is identified as the main problem with the chosen basis, while, on the other hand, its important advantages are pointed out. An effective interaction obtained by a similarity transformation is proposed for improving the convergence of the diagonalization scheme, and numerical experiments are performed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed.