On the path integral simulation of space-time fractional Schroedinger equation with time independent potentials

TL;DR

This paper introduces a path integral method based on Levy measures for solving space-time fractional Schrödinger equations, enabling efficient numerical simulation of quantum systems with fractional dynamics.

Contribution

It proposes a novel Feynman-Kac path integral approach using CTRW and Pareto distributions to simulate space-time fractional quantum systems more effectively.

Findings

Successfully simulates space-time fractional diffusion with standard convergence rates

Uses Pareto distribution for capturing lowest energy states in quantum systems

Provides an alternative numerical method to fractional calculus for fractional PDEs

Abstract

In this work a Feynman-Kac path integral method based on Levy measure has been proposed for solving the Cauchy problems associated with the space-time fractional Schroedinger equations arising in interacting systems in fractional quantum mechanics. The Continuous Time Random Walk(CTRW) model is used to simulate the underlying Levy process-a generalized Wiener process. Since we are interested to capture the lowest energy state of quantum systems, we use Pareto distribution as opposed to Mittag-Leffler random variables, which are more suitable for finite time. Adopting the CTRW model we have been able to simulate the space-time fractional diffusion process with comparable simplicity and convergence rate as in the case of a standard diffusion. We hope this paves an elegant way to solve space-time diffusion equations numerically through Fractional Feynman-Kac path integral technique as an alternative to fractional calculus.