标题： 一种基于标准拉格朗日-拉盖尔基的正则化拉格朗日网格方法 显示英文标题

标题： A regularized Lagrange-mesh method based on an orthonormal Lagrange-Laguerre basis

摘要： 拉格朗日网格方法是一种近似的变分方法，由于使用了高斯求积公式，因此具有网格计算的形式。尽管该方法在许多问题中使用少量网格点时能提供精确的结果，但势能项中的奇异性可能会大大降低其精度。本文设计了一种新的正则化拉格朗日-拉盖尔网格，基于 \textit{确切地} 个正交拉格朗日函数。将其应用于两个可解的径向势：谐振子势和库仑势。尽管库仑势和离心势存在奇异性，但对于所有偏波，均得到了准确的束缚态能量。对这些结果的分析以及与其他拉格朗日网格计算的比较，得出了一个简单的规则，用于预测在何种情况下奇异性会导致拉格朗日网格计算中显著的精度损失。此外，拉格朗日-拉盖尔网格方法被应用于通过积分关系评估相移。少量的网格点足以提供非常精确的结果。 显示英文摘要

摘要： The Lagrange-mesh method is an approximate variational approach having the form of a mesh calculation because of the use of a Gauss quadrature. Although this method provides accurate results in many problems with small number of mesh points, its accuracy can be strongly reduced by the presence of singularities in the potential term. In this paper, a new regularized Lagrange-Laguerre mesh, based on \textit{exactly} orthonormal Lagrange functions, is devised. It is applied to two solvable radial potentials: the harmonic-oscillator and Coulomb potentials. In spite of the singularities of the Coulomb and centrifugal potentials, accurate bound-state energies are obtained for all partial waves. The analysis of these results and a comparison with other Lagrange-mesh calculations lead to a simple rule to predict in which cases a singularity does induce or not a significant loss of accuracy in Lagrange-mesh calculations. In addition, the Lagrange-Laguerre-mesh approach is applied to the evaluation of phase shifts via integral relations. Small numbers of mesh points suffice to provide very accurate results.