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 $(\partial_t-i\triangle_{\Gamma})^{\alpha},\,\,\alpha=1/2,-1/2,-1,\ldots,$ 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.