應用保結構的ΓQR演算法求解Bethe-Salpeter特徵值問題

Abstract

以Hans Bethe和Edwin Salpeter所命名的Bethe-Salpeter方程，在量子物理、量子化學等學界廣泛地應用於許多問題上。Bethe-Salpeter方程在經過離散化後，會得到一個具有特殊結構的Bethe-Salpeter特徵值問題，在此研究中我們將提出一種保結構的ΓQR演算法來計算Bethe-Salpeter特徵值問題的所有特徵值，且保證在每一步的計算中，我們都不會破壞這樣特殊的結構。此外，我們會在演算法中應用一些特殊的技巧，進而提高ΓQR演算法算出特徵值的速度。最後，提供一些與linear response特徵值問題的比較，以及一些數值結果進而顯現ΓQR演算法的可行性。

In applications, there are many problems can be described by the Bethe-Salpeter equation which is named after Hans Bethe and Edwin Salpeter. After discretizing this equation, we will get a Bethe-Salpeter eigenvalue problem, that is, we compute eigenpair of H=[A B; -conj(B) -conj(A)] with A^{H} = A and B^{T} = B. In this study, we propose a structure-preserving ΓQR algorithm to solve the Bethe-Salpeter eigenvalue problem of H in each step. Moreover, to accelerate the convergence of the proposed ΓQR algorithm, some technologies are applied. Comparisons with the linear response eigenvalue problem are provided, and numerical results are presented to illustrate the feasibility of our algorithm.