Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

TL;DR

This paper demonstrates that exploiting centrosymmetry in matrices derived from the Sinc collocation method significantly reduces computational complexity and improves efficiency in solving singular Sturm-Liouville problems, including Schrödinger equations.

Contribution

The authors show that centrosymmetric matrices allow for a reduction in computational effort by transforming large eigensystems into smaller ones, enhancing the Sinc collocation method's efficiency.

Findings

Reduced eigenvalue computation to 1/(N+1) of components

Significant efficiency gains demonstrated in numerical experiments

Improved accuracy in solving Schrödinger equations with anharmonic potential

Abstract

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 Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.