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Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model

特征向量 滤波器(信号处理) 计算 操作员(生物学) 物理 数学分析 功能(生物学) 数学 算法 量子力学 计算机科学 化学 抑制因子 基因 生物 转录因子 进化生物学 生物化学 计算机视觉
作者
Michael R. Wall,Daniel Neuhauser
出处
期刊:Journal of Chemical Physics [American Institute of Physics]
卷期号:102 (20): 8011-8022 被引量:557
标识
DOI:10.1063/1.468999
摘要

In a previous paper we developed a method, Filter-Diagonalization, for extracting eigenvalues and eigenstates of a given operator at any desired energy range. In essence, the method eliminates correlation between distant eigenstates through a short-time filter while correlations between closely lying states are eliminated by diagonalization. Here we extend Filter-Diagonalization. When used to extract eigenvalues for a given operator H, we show that all eigenvalue information is directly extracted from a short segment of the correlation function C(t)=(ψ(0)‖e−iHt‖ψ(0)), or alternately from a small number of residues (ψ(0)‖Rn(H)‖ψ(0)), where ψ(0) is a random initial function and Rn(H) is any desired polynomial expansion in H. The implications of this feature are twofold. First, in contrast to the previous version the wave packet needs only to be propagated once (to prepare C(t)), and eigenstates at all desired energy windows can then be extracted with negligible extra computation time (and negligible storage requirements). In a simulation presented here, accurate eigenvalues are extracted using propagation times which are only a 0.0041 fraction of the ‘‘natural’’ time, i.e., the time by which the relative phase of the two closest eigenstates reaches 2π. The second and more important feature is that the method is automatically suitable for extracting eigenvalues (or normal modes) using a short-time segment of any signal C(t) which is a sum of (unknown) Fourier components (C(t)=∑ndne−iεnt) regardless of its origin. In addition to its use for determining eigenvalues of known operators, the method may also be utilized to extract normal modes from classical-dynamics simulations, eigenstates from real-time Quantum Monte-Carlo studies, frequencies from experimental optical or electrical signals, or be utilized in any other circumstance where a correlation function or general signal is only known for short times (or expensive to generate at long times).
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