AI 中文总结
本文通过无限带状矩阵和变分方法，提出三种基于平移量子谐振子哈密顿量谱的方法，建立更广泛适用的傅里叶不确定性原理。
AI 中文摘要
本文通过无限带状矩阵，采用变分方法最小化不确定性积，研究傅里叶不确定性原理。分析傅里叶不确定性原理的传统方法严重依赖傅里叶变换背后的海森堡-外尔结构，因此在此设置之外难以推广。我们提供了三种不同的方法来发展基于平移量子谐振子哈密顿量 $H_{a,b}$ 谱的不确定性原理。在厄米-高斯基中，$H_{a,b}$ 具有无限三对角形式，这是分析的核心。一种方法通过解析方法精确求解谱，另一种方法将 $H_{a,b}$ 分解为 $2\ imes 2$ 矩阵的无限和，第三种方法是代数方法，通过与量子谐振子的酉等价来建立谱。本文所采用的方法提供了更广泛的替代途径，适用于其他保持三对角形式的积分变换的其他正则共轭算子。
英文摘要
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.
