On the nonreflecting boundary operators for the general two dimensional Schr\"odinger equation

TL;DR

This paper studies nonreflecting boundary operators for the 2D Schrödinger equation, defining fractional operators and corner conditions to improve boundary treatments, with analysis of stability and uniqueness.

Contribution

It introduces fractional boundary operators and corner conditions for Schrödinger equations, enhancing the pseudo-differential boundary approach.

Findings

Operators of the form $( ext{partial}_t - i riangle_ ext{boundary})^eta$ are defined and analyzed.

Corner conditions for rectangular domains are derived and formulated.

Stability and uniqueness of solutions under these boundary conditions are established.

Abstract

Of the two main objectives we pursue in this paper, the first one consists in the studying operators of the form $(\partial_t-i\triangle_{\Gamma})^{\alpha},\,\,\alpha=1/2,-1/2,-1,\ldots,$ where $\triangle_{\Gamma}$ is the Laplace-Beltrami operator. These operators arise in the context of nonreflecting boundary conditions in the pseudo-differential approach for the general Schr\"odinger 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.