Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem
For researchers in many-body physics and quantum chemistry, this provides a more efficient and stable algorithm for a specific eigenvalue problem arising from Green's function evaluation.
The paper extends the structure-preserving doubling algorithm to solve the discretized Bethe-Salpeter eigenvalue problem, addressing potential breakdowns with a double-Cayley transform or three-recursion remedy. Numerical results demonstrate efficiency and structure preservation.
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.