Efficient $\mathcal{O}(N^2)$ approach to solve the Bethe-Salpeter   equation for excitonic bound states
  Excitonic effects in optical spectra and electron-hole pair excitations are described by solutions of the Bethe-Salpeter equation (BSE) that accounts for the Coulomb interaction of excited electron-hole pairs. Although for the computation of excitonic optical spectra in an extended frequency range efficient methods are available, the determination and analysis of individual exciton states still requires the diagonalization of the electron-hole Hamiltonian $\hat{H}$. We present a numerically efficient approach for the calculation of exciton states with quadratically scaling complexity, which significantly diminishes the computational costs compared to the commonly used cubically scaling direct-diagonalization schemes. The accuracy and performance of this approach is demonstrated by solving the BSE numerically for the Wannier-Mott two-band model in {\bf k} space and the semiconductors MgO and InN. For the convergence with respect to the $\vk$-point sampling a general trend is identified, which can be used to extrapolate converged results for the binding energies of the lowest bound states. 
F. Fuchs, C. R\"odl, A. Schleife, and F. Bechstedt