The Fourier Uncertainty Principle Through the Lens of Infinite Banded Matrices

Abstract

In this paper, we investigate the Fourier uncertainty principle through infinite banded matrices by way of a variational approach to minimizing the uncertainty product. Traditional methods for analyzing the Fourier uncertainty principle rely heavily on the Heisenberg--Weyl structure that underlie the Fourier transform and therefore do not generalize well beyond this setting. We provide three distinct approaches to developing the uncertainty principle based on the spectrum of a shifted quantum harmonic oscillator Hamiltonian, $H_{a,b}$. In the Hermite--Gauss basis, $H_{a,b}$ has an infinite, tridiagonal form which is central to the analysis. One approach exactly solves for the spectrum through analytic methods, another decomposes $H_{a,b}$ into an infinite sum of $2\times 2$ matrices, and the third approach is an algebraic approach to establishing the spectrum via a unitary equivalence with the quantum harmonic oscillator. The approaches taken herein provide alternate avenues that are more broadly applicable to other canonically conjugate operators for other integral transforms which maintain a tridiagonal form.