Structure-Preserving ΓQR and Γ-Lanczos Algorithms for Bethe-Salpeter Eigenvalue Problems
This work addresses the need for structure-preserving algorithms in computational quantum physics, specifically for Bethe-Salpeter eigenvalue problems, but the improvements are incremental.
The paper proposes ΓQR and Γ-Lanczos algorithms for Bethe-Salpeter eigenvalue problems that preserve the special structure of the matrix, ensuring computed eigenvalues and eigenvectors retain similar properties. Numerical results demonstrate the methods' superiority.
To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called ΓQR algorithm and Γ-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial matrix $H=\begin{bmatrix}A & B-\overline{B} & -\overline{A}\end{bmatrix}$, resulting the computed eigenvalues and the associated eigenvectors still hold the properties similar to those of $H$. Theorems are given to demonstrate the validity of the proposed two algorithms in theory. Numerical results are presented to illustrate the superiorities of our methods.