Abstract. In this paper, we propose a spectrally accurate solver for computing the Bogoliubov-de Gennes (BdG) excitations of spin-1 Bose-Einstein condensates (BECs), which are governed by the BdG equations, around the mean-field ground state. The BdG equations are essentially a constrained eigenvalue/eigenfunction system. First, we investigate their analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Second, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free, and the operator-function evaluation is accelerated by a discrete fast Fourier transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis of the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and we investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1–3 spatial dimensions.
Li et al. (Wed,) studied this question.