Multiscale quantum propagation using compact-support wavelets in space and time

Abstract

Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.

Time‐dependent solution of molecular quantum systems using multiwavelet

“…Their solutions reside at the border between stability and instability with respect to the time progression and quite sensitive to the numerical manipulation which violates the reversibility. Naive FDM/FEM methods fail to provide stable solutions and we need more sophisticated methods such as split‐step‐Fourier19, 20, Chebyshev integrator method19, 21, 22, and the FDM/FEM with the Cayely algorithm23–25 in order to solve the TDSE. The same situation is encountered in the numerical solutions using the multiwavelet basis.…”

Multimode wavelet basis calculations via the molecular self-consistent-field plus configuration-interaction method

“…[1][2][3][4][5][6] The long-range interest is that these wavelets might be able to provide a robust and general platform for solving wide classes of multidimensional molecular Schrödinger equations. [1][2][3][4][5][6] The long-range interest is that these wavelets might be able to provide a robust and general platform for solving wide classes of multidimensional molecular Schrödinger equations.…”

Additions to the class of symmetric-antisymmetric multiwavelets: Derivation and use as quantum basis functions

“…In the last few years, wavelet applications to quantummechanical problems have begun to be explored in earnest ͑see for example, recent work [1][2][3][4] and references therein͒. Wavelets provide general multiscale flexibility in function approximation, allowing for customizable resolution that is of considerable advantage for functions with large dynamic variation and/or localized features.…”

“…The first of these is the accurate evaluation of projection integrals needed for wave function expansion in wavelet bases and of matrix elements needed for operator expansion. 3 This problem has been addressed by defining new edge wavelets for each family which are able to terminate exactly at required domain edges. 7,8,22,23 The second issue is that of solving boundary or initial value problems.…”