Solution of Cartesian and Curvilinear quantum equations via multiwavelets on the interval

Abstract It is shown how orthogonal compact-support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions. Special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families. All of the integrals required for Hamiltonian matrix elements, involving both regular and edge functions, are calculated efficiently through use of recursion and quadrature methods. It is demonstrated through accurate eigenvalue determination that both Cartesian and curvilinear degrees of freedom are readily accommodated with such a basis, using as examples the particle in a box and the hydrogen atom in spherical polar coordinates.

[1]  John C. Slater,et al.  Wave Functions in a Periodic Potential , 1937 .

[2]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[3]  F. Smith.  Generalized Angular Momentum in Many-Body Collisions , 1960 .

[4]  Electronics Letters , 1965, Nature.

[5]  Bruce R. Johnson,et al.  Adiabatic separations of stretching and bending vibrations: Application to H2O , 1986 .

[6]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[7]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  A. Latto,et al.  Compactly supported wavelets and the numerical solution of Burgers' equation , 1990 .

[9]  Jacques Liandrat,et al.  Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation , 1990 .

[10]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[11]  Y. Meyer.  Ondelettes sur l'intervalle. , 1991 .

[12]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[13]  B. M. Fulk.  MATH , 1992 .

[14]  Using wavelets to solve the Burgers equation: A comparative study. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[15]  Jinchao Xu,et al.  Galerkin-wavelet methods for two-point boundary value problems , 1992 .

[16]  C. Micchelli,et al.  Using the refinement equation for evaluating integrals of wavelets , 1993 .

[17]  M. Defranceschi,et al.  Looking at atomic orbitals through fourier and wavelet transforms , 1993 .

[18]  Cho,et al.  Wavelets in electronic structure calculations. , 1993, Physical review letters.

[19]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[20]  P. Auscher,et al.  Compactly Supported Wavelets and Boundary Conditions , 1993 .

[21]  I. Daubechies,et al.  Multiresolution analysis, wavelets and fast algorithms on an interval , 1993 .

[22]  Sam Qian,et al.  Wavelets and the Numerical Solution of Partial Differential Equations , 1993 .

[23]  Langis Gagnon,et al.  Symmetric Daubechies' wavelets and numerical solution of NLS equations , 1994 .

[24]  Ondelettes sur l'intervalle pour la prise en compte de conditions aux limites , 1995 .

[25]  Juan M. Restrepo,et al.  Wavelet-Galerkin Discretization of Hyperbolic Equations , 1995 .

[26]  Xiaodong Zhou,et al.  Wavelet solutions for the Dirichlet problem , 1995 .

[27]  C. Hwang,et al.  THE COMPUTATION OF WAVELET‐GALERKIN APPROXIMATION ON A BOUNDED INTERVAL , 1996 .

[28]  O. Vasilyev,et al.  A Dynamically Adaptive Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain , 1996 .

[29]  C. Chui,et al.  A study of orthonormal multi-wavelets , 1996 .

[30]  A. Kurdila,et al.  On the conditioning of numerical boundary measures in wavelet galerkin methods , 1996 .

[31]  Xiang-Gen Xia,et al.  Design of prefilters for discrete multiwavelet transforms , 1996, IEEE Trans. Signal Process..

[32]  Wei,et al.  Wavelets in self-consistent electronic structure calculations. , 1996, Physical review letters.

[33]  Roland Glowinski,et al.  Wavelet and Finite Element Solutions for the Neumann Problem Using Fictitious Domains , 1996 .

[34]  Bruce R. Johnson,et al.  Wavelet based in eigenvalue problems in quantum mechanics , 1996 .

[35]  Christoph Schwab,et al.  Fully Discrete Multiscale Galerkin BEM , 1997 .

[36]  C. J. Tymczak,et al.  Orthonormal Wavelet Bases for Quantum Molecular Dynamics , 1997 .

[37]  Xiang-Gen Xia,et al.  A new prefilter design for discrete multiwavelet transforms , 1997, Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers (Cat. No.97CB36136).

[38]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[39]  O. Vasilyev,et al.  A Fast Adaptive Wavelet Collocation Algorithm for Multidimensional PDEs , 1997 .

[40]  S. Goedecker,et al.  Linear scaling solution of the coulomb problem using wavelets , 1997, physics/9701024.

[41]  Christian Lage,et al.  Wavelet Galerkin algorithms for boundary integral equations , 1997 .

[42]  Wolfgang Dahmen,et al.  Multiscale Wavelet Methods for Partial Differential Equations , 1997 .

[43]  James T. Miller,et al.  Adaptive multiwavelet initialization , 1998, IEEE Trans. Signal Process..

[44]  S. Goedecker.  Wavelets and Their Application: For the Solution of Partial Differential Equations in Physics , 1998 .

[45]  D. Hardin,et al.  Multiwavelet prefilters. 1. Orthogonal prefilters preserving approximation order p/spl les/2 , 1998 .

[46]  Donald J. Kouri,et al.  Shannon–Gabor wavelet distributed approximating functional , 1998 .

[47]  H. L. Resnikoff,et al.  Wavelet analysis: the scalable structure of information , 1998 .

[48]  Pascal Monasse,et al.  Orthonormal wavelet bases adapted for partial differential equations with boundary conditions , 1998 .

[49]  Akram Aldroubi,et al.  Projection based prefiltering for multiwavelet transforms , 1998, IEEE Trans. Signal Process..

[50]  Patrick Fischer,et al.  Numerical Solution of the Schrödinger Equation in a Wavelet Basis for Hydrogen-like Atoms , 1998 .

[51]  D. Hoffman,et al.  An application of distributed approximating functional-wavelets to reactive scattering , 1998 .

[52]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[53]  Alan Edelman,et al.  Multiscale Computation with Interpolating Wavelets , 1998 .

[54]  A. Diaz.  A wavelet-Galerkin scheme for analysis of large-scale problems on simple domains , 1999 .

[55]  Stefan Goedecker,et al.  FREQUENCY LOCALIZATION PROPERTIES OF THE DENSITY MATRIX AND ITS RESULTING HYPERSPARSITY IN A WAVELET REPRESENTATION , 1999 .

[56]  T. Arias.  Multiresolution analysis of electronic structure: semicardinal and wavelet bases , 1998, cond-mat/9805262.

[57]  Peter N. Heller,et al.  The application of multiwavelet filterbanks to image processing , 1999, IEEE Trans. Image Process..

[58]  Jiao Licheng,et al.  Adaptive multiwavelet prefilter , 1999 .

[59]  J. L. Kinsey,et al.  Quadrature integration for orthogonal wavelet systems , 1999 .

[60]  Kyeongjae Cho,et al.  Wavelets in all-electron density-functional calculations , 1999 .

[61]  Bruce R. Johnson,et al.  Quadrature prefilters for the discrete wavelet transform , 2000, IEEE Trans. Signal Process..

[62]  R. DeVore,et al.  Multiscale decompositions on bounded domains , 2000 .

[63]  Numerical solution of eigenvalue problems by means of a wavelet‐based Lanczos decomposition , 2000 .

[64]  Bruce R. Johnson,et al.  Multiwavelet moments and projection prefilters , 2000, IEEE Trans. Signal Process..

[65]  W. Dahmen,et al.  Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines , 2000 .

[66]  N. B. Skachkov,et al.  On the application of , 2002 .

[67]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[68]  G. Beylkin,et al.  On the representation of operators in bases of compactly supported wavelets , 1992 .