Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

Abstract

We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.

Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

“…For comparison of the performance and effectiveness of different PMLs and/or ABCs for NLSE/GPE and their applications, we refer to [17,135,186,187,189] and references therein.…”

Reflectionless propagation of Manakov solitons on a line:A model based on the concept of transparent boundary conditions

“…One of the effective approaches to solve such a problem can be done in the framework of the concept of transparent boundary conditions. Such transparent boundary conditions for evolution equations can be constructed by coupling the solutions of the initial value boundary problems (IVBPs) in the interior and exterior domains [25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Transparent vertex boundary conditions for quantum graphs: Simplified approach

“…4. Contour plot of the probability density for the GWP initially located on the first bond and given by (14). The imposed vertex boundary conditions are Kirchhoff-type ones with α 1 = 2. two identical fractions.…”

“…The problem of transparent boundary conditions (TBC) for linear partial differential equations (PDEs) was first introduced in the Refs. [1,2] and has become a well developed topic in mathematical and theoretical physics now (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] for review). For both (absorbing and transparent transmission) processes, the boundary conditions can be derived by factorization of the differential operator corresponding to a wave equation, which, in general, leads to complicated equations for the boundary conditions.…”