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

V. A. Derkach, C. Trunk, J. R. Yusupov, D. U. Matrasulov

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.

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

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.

Paper Structure

This paper contains 8 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries. Sturm-Liouville operators and Weyl-Titchmarsh coefficients
  3. Transparent boundary conditions for discrete Schrödinger operators
  4. General case
  5. Examples
  6. Harmonic potential
  7. Pöschl-Teller potential
  8. Conclusions

Key Result

Theorem 3

Let the differential expression $\mathcal{A}=-\frac{d^2}{dx^2}+V$ satisfy (H1) and (H2), let $(a_-,a_+)$ be a finite interval in $\mathbb{R}$, and let $m_-$ and $m_+$ be the Weyl-Titchmarsh coefficients for $A_\pm$, cf. eTWeyl. Then transparent boundary conditions for $\mathcal{A}$ on the interval $

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Corollary 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Theorem 8
  • Corollary 9
  • Corollary 10
  • ...and 3 more