Orthogonal fast spherical Bessel transform on uniform grid

TL;DR

This paper introduces an efficient algorithm for the orthogonal fast discrete spherical Bessel transform on uniform grids, enabling improved numerical solutions of 3D Schrödinger equations through a novel transform factorization.

Contribution

It presents a new algorithm that combines Fourier and Legendre polynomial transforms for the spherical Bessel transform, enhancing computational efficiency.

Findings

Demonstrated effectiveness in solving 3D time-dependent Schrödinger equation

Achieved faster and more accurate numerical results

Validated the method's utility through practical implementation

Abstract

We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on an uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast Fourier transform and the orthogonal transform founded on the derivatives of the discrete Legendre orthogonal polynomials. The method utility is illustrated by its implementation for the numerical solution of the three-dimensional time-dependent Schr\"odinger equation.