特征向量延拓的收敛性
Convergence of Eigenvector Continuation
摘要：
特征向量延拓（英语：Eigenvector continuation）是一种计算方法，它求出具有一个或多个控制参数的哈密顿矩阵的极值特征值和特征向量。它通过投影到对应于控制参数的选定训练值的特征向量的子空间上来实现这一点。该方法已被证明是非常有效和准确的内插和外推特征向量。然而，几乎没有人知道该方法是如何收敛的，其快速收敛特性仍然是个谜。本文首次研究了特征向量延拓的收敛性。为了进行数学分析，我们引入了一个新的变种的特征向量延拓，我们称之为向量延拓。首先证明了特征向量延拓与向量延拓具有相同的收敛性质，然后分析了向量延拓的收敛性。我们的分析表明，一般来说，特征向量延拓比扰动理论收敛得更快。更快的收敛是通过消除一种我们称之为微分折叠的现象来实现的，微分折叠是微扰理论中出现在不同阶的非正交向量之间的干扰。从我们的分析中，我们可以预测如何本征向量延拓收敛内和外的微扰理论的收敛半径。而特征向量延拓是一种非微扰方法，我们表明，它的收敛速度可以推导出从幂级数展开的特征向量。我们的结果也产生新的见解的性质分歧微扰理论。
英文摘要：
Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to selected training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. However, almost nothing is known about how the method converges, and its rapid convergence properties have remained mysterious. In this letter we present the first study of the convergence of eigenvector continuation. In order to perform the mathematical analysis, we introduce a new variant of eigenvector continuation that we call vector continuation. We first prove that eigenvector continuation and vector continuation have identical convergence properties and then analyze the convergence of vector continuation. Our analysis shows that, in general, eigenvector continuation converges more rapidly than perturbation theory. The faster convergence is achieved by eliminating a phenomenon that we call differential folding, the interference between non-orthogonal vectors appearing at different orders in perturbation theory. From our analysis we can predict how eigenvector continuation converges both inside and outside the radius of convergence of perturbation theory. While eigenvector continuation is a non-perturbative method, we show that its rate of convergence can be deduced from power series expansions of the eigenvectors. Our results also yield new insights into the nature of divergences in perturbation theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
