Convergence of a moving window method for the Schrödinger equation with potential on $\mathbb{R}^d$

TL;DR

The paper tackles the challenge of solving the linear Schrödinger equation with a spatially localized potential on $\mathbb{R}^d$ by introducing a moving window approach that truncates to a bounded, periodically forced domain $\Omega_L=[-L,L]^d$ using a smooth cutoff $\chi_L$. The method recasts the problem on the scaled torus and applies a first-order exponential integrator with Fourier discretization, while providing rigorous error bounds that relate the whole-space solution to the truncated, zero-extended numerical solution. For initial data in $H^γ(\mathbb{R}^d)\cap L^2(\mathbb{R}^d;|x|^{2γ}dx)$ with $γ\ge2$, the scheme achieves first-order convergence in time and $γ/2$-order convergence in space; a CFL condition yields half-order convergence for $γ=1$. A key feature is a dynamic domain-extension strategy to mitigate reflections, enabling long-time simulations. The results are supported by extensive numerical experiments on free propagation, quantum tunneling, and lattice scattering, demonstrating the practicality and accuracy of the moving window framework for unbounded-domain dispersive problems.

Abstract

We propose a novel framework, called moving window method, for solving the linear Schrödinger equation with an external potential in $\mathbb{R}^d$. This method employs a smooth cut-off function to truncate the equation from Cauchy boundary conditions in the whole space to a bounded window of scaled torus, which is itself moving with the solution. This allows for the application of established schemes on this scaled torus to design algorithms for the whole-space problem. Rigorous analysis of the error in approximating the whole-space solution by numerical solutions on a bounded window is established. Additionally, analytical tools for periodic cases are used to rigorously estimate the error of these whole-space algorithms. By integrating the proposed framework with a classical first-order exponential integrator on the scaled torus, we demonstrate that the proposed scheme achieves first-order convergence in time and $γ/2$-order convergence in space for initial data in $H^γ(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2γ} dx)$ with $γ\geq 2$. In the case where $γ= 1$, the numerical scheme is shown to have half-order convergence under an additional CFL condition. In practice, we can dynamically adjust the window when waves reach its boundary, allowing for continued computation beyond the initial window. Extensive numerical examples are presented to support the theoretical analysis and demonstrate the effectiveness of the proposed method.

Figures (9)

• Figure 1: Modulus of the exact and numerical solutions (free Schrödinger equation).
• Figure 2: Errors of the numerical solutions at $T = 1$ (free Schrödinger equation).
• Figure 3: Comparison between exact and numerical solutions (free Schrödinger equation).
• Figure 4: Numerical solution of $|u|$ with Type I initial function (quantum tunneling example).
• Figure 5: Convergence rate of numerical solution at $T = 1$ (quantum tunneling example).

Theorems & Definitions (20)

• Theorem 2.1
• Remark 2.1
• Lemma 3.1: Global well-posedness
• Lemma 3.2: Bernstein's inequalities
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5: Approximation result and Bernstein's inequality on $\Omega_L$
• proof
• Remark 3.1