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

Abstract

Abstract. We consider dimension reduction for the three-dimensional (3D) Schrödinger equation with the Coulomb interaction and an anisotropic confining potential to lower-dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model (SAM) or surface density model (SDM) in two dimensions (2D) and the line adiabatic model (LAM) in one dimension (1D). Efficient and accurate numerical methods for computing ground states and dynamics of the SAM, SDM, and LAM models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower-dimensional models. They are applied to find numerically convergence and convergence rates for the dimension reduction from 3D to 2D and 3D to 1D in terms of ground state and dynamics, which confirm some existing analytical results for the dimension reduction in the literature. In particular, we explain and demonstrate that the standard Schrödinger-Poisson system in 2D is not appropriate to simulate a "2D electron gas" of point particles confined to a plane (or, more generally, a 2D manifold), whereas SDM should be the correct model to be used for describing the Coulomb interaction in 2D in which the square root of Laplacian operator is used instead of the Laplacian operator. Finally, we report ground states and dynamics of the SAM and SDM in 2D and LAM in 1D under different setups.

Accurate and Efficient Numerical Methods for Computing Ground States and Dynamics of Dipolar Bose-Einstein Condensates via the Nonuniform FFT

“…respectively. Moreover, the chemical potential can be reformulated as 1) is taken as the harmonic potential (1.4) [7,15,16,36], the energies of the ground state satisfy the following virial identity…”

Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT

“…On the other hand, the Coulomb interaction kernel U (x) in 2D is the Green's function of the square-root-Laplace operator instead of the Laplace operator and thus the nonlocal Coulomb interaction ϕ in (1.2) also satisfies the fractional Poisson equation in 2D √ −∆ ϕ(x, t) = |ψ(x, t)| 2 , x ∈ R 2 , lim |x|→∞ ϕ(x, t) = 0, t ≥ 0. (1.8) In this case, (1.1)-(1.2) could be obtained from the 3D SPS under an infinitely strong external confinement in the z-direction [9,14]. This model could be used for modelling 2D materials such as graphene and "electron sheets" [19].…”

Wellposedness of Variable-Coefficient Conservative Fractional Elliptic Differential Equations

“…The fractional Laplacian operator (−∆u) s of order 0 < s < 1 has been used to model nonlocal behavior in many physical problems [2,4,12,21] and has appeared as the infinitesimal generator of a stable Lévy process [2,17,18,38]. (−∆) s can be defined as a pseudodifferential operator of symbol |ξ| 2s on the entire space R d [2] (−∆) s u = F −1 |ξ| 2s F u(ξ) , ∀u ∈ S (1.1) equivalently be defined by the prescription [27] (−∆) s u(x) = C(d, s) P.V.…”