Universal Lattice Basis

TL;DR

This paper introduces a Universal Lattice Basis derived from Shannon's Sampling theorem, offering superior interpolation, orthonormality, and efficient derivatives for quantum systems, with demonstrated convergence and equivalence in bounded and periodic domains.

Contribution

It develops a novel basis based on Shannon's theorem with superior properties for quantum mechanical computations, extending to periodic and bounded domains.

Findings

Superior interpolation properties compared to traditional methods

Quadratic convergence for band-limited functions like Gaussians

Equivalence to Fourier Transform-based spaces

Abstract

We report on the utility of using Shannons Sampling theorem to solve Quantum Mechanical systems. We show that by extending the logic of Shannons interpolation theorem we can define a Universal Lattice Basis, which has superior interpolating properties compared to traditional methods. This basis is orthonormal, semi-local, has a Euclidean norm, and a simple analytic expression for the derivatives. Additionally, we can define a bounded domain for which band-limited functions, such as Gaussians, show quadratic convergence in the representation error in respect to the sampling frequency. This theory also extends to the periodic domain and we illustrate the simple analytic forms of the periodic semi-local basis and derivatives. Additionally, we show that this periodic basis is equivalent to the space defined by the Fast Fourier Transform. This novel basis has great utility in solving quantum mechanical problems for which the wave functions are known to be naturally band-limited. Several numerical examples in single and multi-dimensions are given to show the convergence and equivalence of the periodic and bounded domains for compact states.