Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

Abstract

ABSTRACT:The simple Lanczos process is very efficient for finding a few extreme eigenvalues of a large symmetric matrix. The main task in each iteration step consists in evaluating a matrix-vector product. It is shown how to apply a fast wavelet-based product in order to speed up computations. Some numerical results are given for three different monodimensional cases: the harmonic oscillator case, the hydrogenlike atoms, and a problem with a pseudo-double-well potential.

Compactly Supported Wannier Functions and Strictly Local Projectors

“…An extreme form of localization known as compact support or strict locality has also been studied in a variety of contexts. Compactly supported orthogonal wavelets [15,16], a popular tool from signal processing, have been used to understand a variety of quantum physics problems [17,18,19]. In the limit of zero potential, Wannier functions are known to be a specific form of wavelets [20].…”

Wavelets in curvilinear coordinate quantum calculations: H2+ electronic states

“…This advantage has great appeal for problems exhibiting large dynamic ranges. A number of papers have investigated wavelet use in quantum mechanical problems, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] though this technology is only in the early stages of development. In the case of orthogonal compact support wavelets, for instance, it is only very recently that it has been shown possible to solve the standard hydrogen atom problem to more than a few decimal places.…”