跃迁振幅对角化对量子系统性质的影响I：离散化效应
Properties of Quantum Systems via Diagonalization of Transition Amplitudes I: Discretization Effects
摘要：
分析了基于空间离散短时演化算符精确对角化的非相对论量子系统性质的计算方法。在本文中，我们提出了一个详细的分析与空间离散化的错误。使用实空间离散化哈密顿量的直接对角化的方法导致离散化间距$Δ$中的多项式误差。在这里，我们表明，该方法的基础上的对角化的短时间演化算子导致小得多的离散化误差，消失指数与$1/Δ^2$。因此，所提出的计算方案特别适合于少体量子系统的数值研究。解析推导的离散化误差估计的数值结果表明，几个模型。在后续的论文[1]中，我们提出并分析了这种方法与最近引入的用于高精度计算短时传播的有效作用方案合并所带来的实质性改进。
英文摘要：
We analyze the method for calculation of properties of non-relativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors associated with space discretization. Approaches using direct diagonalization of real-space discretized Hamiltonians lead to polynomial errors in discretization spacing $Δ$. Here we show that the method based on the diagonalization of the short-time evolution operators leads to substantially smaller discretization errors, vanishing exponentially with $1/Δ^2$. As a result, the presented calculation scheme is particularly well suited for numerical studies of few-body quantum systems. The analytically derived discretization errors estimates are numerically shown to hold for several models. In the followup paper [1] we present and analyze substantial improvements that result from the merger of this approach with the recently introduced effective-action scheme for high-precision calculation of short-time propagation.
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