On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system

Dimension Reduction of the Schrödinger Equation with Coulomb and Anisotropic Confining Potentials

“…Thus the system of (1.1)-(1.2) is usually called the Schrödinger-Poisson system (SPS) in the literature [12,50]. In fact, the corresponding rigorous derivation of this kind of "Hartree equations" was started from a Hartree ansatz for the many-body (e.g., N -body) wave-function by using a "weak coupling scaling" (i.e., a factor 1/N in front of the Coulomb interaction potential) and passing to the limit N → ∞ in the BBGKY hierarchy [13,14,29].…”

“…In fact, the spatial confinement is an essential feature of many "nanoscale devices" and has gained much attention from both experimental and mathematical studies [3,31,40,42]. Although the SPS (1.3)-(1.4) in 2D or 1D has been used in some of the literature [1,3,12,25,27,42,43,48,50] to simulate low-dimensional quantum systems of fermions such as 2D "electron sheets" or 1D "quantum wires," it is highly debated or mathematically mysterious whether the above SPS is an appropriate model for these confining low-dimensional quantum systems. In fact, intuitively point particles confined to a 2D manifold still interact with the Coulomb interaction potential at O 1 |x| in 2D; thus it seems that the SPS (1.3)-(1.4) in 2D is not an appropriate model.…”

“…In order to verify numerically the dimension reduction from the 3D SPS (1.1)-(1.2) to the 2D SAM (2.10)-(2.11) with (2.19) and SDM (2.16)-(2.17) and 1D LAM (2.30)-(2.31) with (2.36), to find numerically the convergence rates for the dimension reduction, and to simulate numerically low-dimensional quantum systems based on the 2D and 1D models, in this section, we briefly introduce numerical methods for computing ground states and dynamics of the 2D SAM and SDM and 1D LAM as well as 2D SPS models. For efficient and accurate numerical methods for computing ground states and dynamics of the SPS (1.3)-(1.4) in 3D and 1D, we refer the reader to [26,50] and references therein. In practical computations, the whole space problems (2.…”

Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars

“…In order to design numerical methods for computing the ground state, we first construct a gradient flow with discrete normalization (GFDN) which was widely and successfully used in computing ground states of Bose-Einstein condensation [5,3,21] and the Schrödinger-Poisson-Slater equations [36]. Then the problem is truncated into a box with homogeneous Dirichlet boundary conditions and a backward Euler sine pseudospectral method [4,3,36] is applied to discretize it.…”

“…Then the problem is truncated into a box with homogeneous Dirichlet boundary conditions and a backward Euler sine pseudospectral method [4,3,36] is applied to discretize it. For computing the dynamics, again the problem is truncated into a box with homogeneous Dirichlet boundary conditions and a timesplitting sine pseudospectral method [6,7,9,3,36] is applied to discretize it. In particular, when the potential and initial data for dynamics are spherically symmetric, then the problem collapses to a quasi-1D problem.…”