Transparent boundary conditions for the stationary Schroedinger equation via Weyl-Titchmarsh theory

TL;DR

This paper introduces a general method for deriving transparent boundary conditions for the stationary Schrödinger equation using Weyl-Titchmarsh theory, applicable to arbitrary potentials, with specific examples for harmonic and Pöschl-Teller potentials.

Contribution

The paper presents a novel approach to formulate transparent boundary conditions for the stationary Schrödinger equation utilizing Weyl-Titchmarsh coefficients, broadening applicability to various potentials.

Findings

Boundary conditions expressed via Weyl-Titchmarsh coefficients.

Application demonstrated for harmonic potential.

Application demonstrated for Pöschl-Teller potential.

Abstract

We propose a general approach for deriving transparent boundary conditions for the stationary Schroedinger equation with arbitrary potential. It is proven that the transparent boundary conditions can be written in terms of the Weyl-Titchmarsh coefficients. As examples for the application of the proposed approach, two special cases for the stationary Schroedinger equation with the harmonic potential and the Poeschl-Teller potential are considered.