Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem

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.

22 pages

References in corpus (3)

• Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem
• A reduced basis approach for calculation of the Bethe-Salpeter excitation energies using low-rank tensor factorizations
• Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation