Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem

Guo, Zhen-Chen; Chu, Eric; Lin, Wen-Wei

doi:10.1090/mcom/3398

Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem
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by Zhen-Chen Guo, Eric King-Wah Chu and Wen-Wei Lin HTML | PDF

Math. Comp. 88 (2019), 2325-2350 Request permission

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 the structure-preserving nature of the algorithm.

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