在Sturm-Liouville问题中利用正弦摄动法的中心对称矩阵
Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems
AI总结：
本文提出利用中心对称矩阵特性减少奇异Sturm-Liouville问题特征值计算复杂度，通过将大特征系统分解为两个小系统，仅需存储1/(N+1)的元素即可获得所有特征值。
摘要：
最近，我们利用双指数变换的正弦摄动法计算了奇异Sturm-Liouville问题的特征值。在本文中，我们证明当算子与奇偶算子对易时，此类微分特征值问题的特征值计算复杂度可显著降低。此时，由正弦摄动法得到的矩阵为中心对称矩阵。利用已知的中心对称矩阵性质，我们将解决一个大特征系统的问题转化为解决两个小特征系统的问题。我们证明仅需计算并存储所有元素中的1/(N+1)即可获得所有特征值，其中(2N+1)对应特征系统的维度。我们将其结果应用于具有非谐振子势的薛定谔方程，并在数值结果部分清晰地展示了使用所提算法在效率和准确性上的显著提升。
英文摘要：
Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schrödinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.
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。
