A time‐independent approach for computing wave functions of the Schrödinger–Poisson system

Abstract

SUMMARYWe describe a two-grid finite element discretization scheme for computing wave functions of the Schrödinger-Poisson (SP) system. To begin with, we compute the first k eigenpairs of the Schrödinger-Poisson eigenvalue (ESP) problem on the coarse grid using a continuation algorithm, where the nonlinear Poisson equation is solved iteratively. We use the k eigenpairs obtained on the coarse grid as initial guesses for computing their counterparts of the ESP on the fine grid. The wave functions of the SP system can be easily obtained using the formula of separation of variables. The proposed algorithm has the following advantages. (i) The initial approximate eigenpairs used in the fine grid can be obtained with low computational cost. (ii) It is unnecessary to discretize the partial derivative of the wave function with respect to the time variable in the SP system. (iii) The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed. Numerical results on the ESP and the SP system are reported. In particular, the rate of convergence of the proposed algorithm is O(h 4 ).

Spectral collocation and a two-level continuation scheme for dipolar Bose–Einstein condensates

“…Table 1: Different quantities of the ground state solution of dipolar BEC obtained using Algorithm 3.1 (first row) and the backward Euler sine pseudospectral method (second row), where V (x, y, z) = (x 2 + y 2 + 0.25z 2 )/2, n = (0, 0, 1) T , and β = 207. 16. (a) The harmonic potential V (x, y, z) = ( 3: The first bifurcation points and associated energy levels of (1.11) with periodic potential V (x, y, z) = (x 2 + y 2 + z 2 )/2 + 100(sin 2 (πx/2) + sin 2 (πy/2) + sin 2 (πz/2)), where n = (0, 0, 1) T , β = 401.432, and λ = 64.22912.…”

“…The numerical study for the self-consistent SP eigenvalue problem can be found in [15]. Recently, Chien et al [16] described a two-grid finite element discretization scheme, which is a time-independent approach for computing wave functions of the self-consistent SP system.…”

Numerical Continuation for Nonlinear Schrödinger Equations

“…See e.g. [Chang et al, 2007a[Chang et al, , 2007bChien et al, 2007]. In conclusion, we can exploit the two-grid discretization schemes to trace solution curves of Eq.…”