A high-order Caputo-based numerical scheme for approximating the one-dimensional time-independent fractional Schrödinger equation

Abstract

The time-independent fractional Schrödinger equation is pivotal for modeling quantum systems exhibiting nonlocal interactions and anomalous dispersion, for which analytical solutions are often intractable. This paper presents a high-order numerical scheme to approximate the solution of the one-dimensional time-independent Schrödinger equation incorporating Caputo fractional derivatives of order α, comprehensively covering the ranges 0 < α ≤ 1 and 1 < α ≤ 2. The core of our approach lies in the development of novel finite difference discretizations. For 0 < α ≤ 1, we employ a second-order weighted-shifted Grünwald approximation to achieve higher accuracy than standard schemes. For 1 < α ≤ 2, a distinct high-order approximation is derived, which is further extended to a fourth-order scheme for enhanced precision. A rigorous theoretical analysis is provided, including a detailed error bound proof that establishes the scheme’s accuracy and stability. We also investigate the convergence rate and the impact of numerical errors on long-term simulations, demonstrating the method’s robustness. The validity and efficiency of the proposed method are confirmed through several numerical examples. The results show excellent agreement with exact solutions in limiting cases (e.g., α → 2) and confirm that the observed convergence orders align with our theoretical predictions. The findings indicate that the developed high-order Caputo-based scheme is a powerful and reliable tool for solving fractional Schrödinger equations, offering significant improvements in accuracy for a wide range of fractional orders.