On the nonreflecting boundary operators for the general two dimensional Schrödinger equation

Abstract

Of the two main objectives we pursue in this paper, the first one consists in the studying operators of the form (∂ t −i Γ ) α , α = 1/2, −1/2, −1, . . . , where Γ is the Laplace-Beltrami operator. These operators arise in the context of nonreflecting boundary conditions in the pseudo-differential approach for the general Schrödinger equation. The definition of such operators is discussed in various settings and a formulation in terms of fractional operators is provided. The second objective consists in deriving corner conditions for a rectangular domain in order to make such domains amenable to the pseudo-differential approach. Stability and uniqueness of the solution is investigated for each of these novel boundary conditions.

Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation with a Vector Potential

“…A similar approach for general domains was introduced in [11], but the conditions are given in the Laplace transform domain, and efficiently recovering time domain conditions is not straightforward. Halfspace conditions were derived by a different method in [12,13,14], and used to assemble conditions for a rectangle in d = 2 and a box in d = 3. These are written in terms of auxiliary unknowns, obtained by solving lower-dimensional Schrödinger equations on boundary faces and edges.…”