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

We consider dimension reduction for the three-dimensional (3D) Schrodinger 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 Schrodinger--Poisson system in 2...

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