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 nonorthogonal 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 nonperturbative 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.

中文翻译：

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特征向量连续收敛

特征向量连续是一种计算方法，用于查找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。它是通过投影到特征向量的子空间上来实现的，该特征向量对应于控制参数的选定训练值。已经证明该方法对于内插和外推特征向量非常有效且准确。但是，关于该方法如何收敛几乎一无所知，其快速收敛特性仍然是个谜。在这封信中，我们提出了特征向量连续性收敛的第一个研究。为了进行数学分析，我们引入了特征向量连续性的新变种，我们称之为向量连续性。我们首先证明特征向量连续和向量连续具有相同的收敛性，然后分析向量连续的收敛。我们的分析表明，总的来说，特征向量连续性的收敛速度比扰动理论要快。通过消除一种称为微分折叠的现象（在微扰理论中以不同顺序出现的非正交矢量之间的干扰）可以实现更快的收敛。从我们的分析中，我们可以预测本征矢量连续性如何在扰动理论的收敛半径内外收敛。虽然本征向量连续是一种非微扰方法，但我们表明其收敛速度可以从本征向量的幂级数展开中推导。