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.
翻译:一种计算方法,它发现汉密尔顿矩阵中带有一个或多个控制参数的极限成份值和成份值。它通过向一个与控制参数中选定的培训值相对应的叶质数子子空间投影来达到这个结果。该方法已证明非常高效和准确,可以对内推和外推电源值进行计算。然而,对于该方法如何趋同及其快速趋同特性仍然神秘地存在,几乎不知道该方法是如何趋同的。在这封信中,我们首次对叶质变异性延续的趋同性进行了剖析。为了进行数学分析,我们还引入了一种新变异的叶质变异性结果,我们称之为矢量延续。我们首先证明,树源的延续性和矢量的延续性具有相同的趋同性,然后分析矢量继续的趋同性。我们的分析显示,一般来说,叶质变异性延续比扰动理论更迅速趋同。通过消除一种我们称之为折叠化的现象,我们称非分子变异性矢量矢量的延续力之间的干扰,我们从外部的理论中可以预测其内部趋同性持续的理论。