Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem

TL;DR

This paper introduces a structure-preserving doubling algorithm for solving the discretized Bethe-Salpeter eigenvalue problem, with convergence analysis and techniques to avoid breakdowns, demonstrating efficiency through numerical tests.

Contribution

It extends the doubling algorithm to the Bethe-Salpeter eigenvalue problem and analyzes its convergence and stability, including remedies for potential breakdowns.

Findings

The algorithm effectively preserves problem structure.

It avoids breakdowns using double-Cayley transform or three-recursion.

Numerical results confirm efficiency and stability.

Abstract

The discretized Bethe-Salpeter eigenvalue problem arises in the Green's function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for $H \in \mathbb{C}^{2n\times 2n}$ with a Hamiltonian-like structure. After an appropriate transformation of $H$ to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and structure-preserving nature of the algorithm.