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The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik

TL;DR

The paper addresses the numerical solution of a generalized 2D time-dependent Schrödinger equation on a semi-infinite strip with variable coefficients and nonreflecting boundary conditions. It develops a Strang-type splitting in the potential for a Crank-Nicolson scheme equipped with discrete transparent boundary conditions, yielding a stable, mass-conserving method that can be implemented efficiently via FFT in the transverse direction. The discrete TBC is constructed as a time-convolution operator $\mathcal{S}_{\rm ref}$, leveraging a transverse eigenbasis $\{E_l,\lambda_{l\delta}\}$ and a $y$-direction FFT to achieve decoupled 1D problems with favorable complexity. Numerical experiments on rectangular barriers demonstrate clear tunnel effects, absence of spurious reflections, and favorable convergence and stability properties across the tested scenarios.

Abstract

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.

The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

TL;DR

The paper addresses the numerical solution of a generalized 2D time-dependent Schrödinger equation on a semi-infinite strip with variable coefficients and nonreflecting boundary conditions. It develops a Strang-type splitting in the potential for a Crank-Nicolson scheme equipped with discrete transparent boundary conditions, yielding a stable, mass-conserving method that can be implemented efficiently via FFT in the transverse direction. The discrete TBC is constructed as a time-convolution operator , leveraging a transverse eigenbasis and a -direction FFT to achieve decoupled 1D problems with favorable complexity. Numerical experiments on rectangular barriers demonstrate clear tunnel effects, absence of spurious reflections, and favorable convergence and stability properties across the tested scenarios.

Abstract

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time -stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.

Paper Structure

This paper contains 4 sections, 5 theorems, 64 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Schrödinger equation on a semi-infinite strip and the splitting in potential Crank-Nicolson scheme with an approximate TBC
  3. The splitting in potential Crank-Nicolson scheme on the infinite space mesh and the discrete TBC
  4. Numerical experiments

Key Result

Proposition 2.1

Let the operator ${\mathcal{S}}$ satisfy an inequality DZ06 for any function $\Phi$: $\overline{\omega}_\delta\times\overline{\omega}^{\,\tau}\to \mathbb{C}$ such that $\Phi^0=0$ and $\left.\Phi\right|_{k=0,K}=0$, where $\boldsymbol\Phi^m=\{\Phi^1,\ldots,\Phi^m\}$. Then, for a solution of the splitting in potential scheme sd11-sd21, the following stability b

Figures (3)

  • Figure 1: The modulus (left) and the real part (right) of the initial function $\psi_G$ together with the normalized barrier from Example B
  • Figure 2: Example B ($Q=1500$). The absolute and relative errors in $C$ and $L^2$ norms in dependence with time for the numerical solution for $(J,K,M)=(1200,64,600)$
  • Figure 3: Example B ($Q=1500$). The modulus and the real part of the numerical solution $\Psi^m$ for $(J,K,M)=(1200,64,600)$

Theorems & Definitions (10)

  • Proposition 2.1
  • proof
  • Corollary 2.1
  • proof
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • Proposition 3.2