[Paper Review] Efficient simulation of Schrödinger equation with piecewise constant positive potential

Xuxin Yang, Antti Rasila+1 more|arXiv (Cornell University)|Dec 4, 2015

Model Reduction and Neural Networks6 references3 citations

TL;DR

This paper introduces the Killing Walk on Spheres (KWOS) algorithm for efficiently simulating weak solutions of the Schrödinger equation with piecewise constant positive potentials. By combining the classical walk on spheres with exponential killing via panharmonic measures, the method enables accurate numerical solutions through stochastic simulation, particularly effective for quantum tunneling and Yukawa-type potentials.

ABSTRACT

In this paper we introduce a new method for the simulation of a weak solution of the Schrödinger-type equation where the potential is piecewise constant and positive. The method, called killing walk on spheres algorithm, combines the classical walk of spheres algorithm with killing that can be determined by using panharmonic measures.

Motivation & Objective

• To develop an efficient numerical method for solving the time-independent Schrödinger equation with discontinuous, piecewise constant positive potentials.
• To address the challenge of simulating weak solutions when classical $ C^2 $ solutions do not exist due to discontinuous potentials.
• To extend the classical walk on spheres algorithm by incorporating exponential killing to model positive potentials via panharmonic measures.
• To provide a computationally feasible simulation framework for problems in quantum tunneling and Yukawa-type equations.
• To validate the method through one- and two-dimensional examples, including mixed Laplace-Yukawa problems.

Proposed method

• The KWOS algorithm combines the classical walk on spheres with exponential killing, where killing rates correspond to the potential values in each subdomain.
• It uses panharmonic measures $ H_{\lambda}^x(D;dy) = \int_0^\infty e^{-\lambda t} h^x(D;dy,t) dt $ to represent solutions of the Yukawa equation with constant $ \lambda $.
• The Radon-Nikodym derivative $ Z_\lambda^x(D;y) = \mathbb{E}^x[e^{-\lambda \tau_D} \mid W_{\tau_D} = y] $ links harmonic and panharmonic measures for efficient simulation.
• In regions with zero potential, the algorithm uses standard harmonic measure (Gambler's ruin) for exit probabilities.
• In regions with positive potential $ \lambda_m $, the KWOS algorithm simulates Brownian motion with exponential killing using recursive sphere-hopping.
• The method avoids direct exit time estimation by interpreting the potential as a killing rate, enabling efficient path simulation via stochastic integration.

Experimental results

Research questions

• RQ1 How can weak solutions of the Schrödinger equation with discontinuous, piecewise constant positive potentials be efficiently simulated?
• RQ2 Can the classical walk on spheres algorithm be extended to handle positive potentials through a killing mechanism?
• RQ3 What is the role of panharmonic measures in representing solutions to the Yukawa equation with constant potentials?
• RQ4 How does the combination of harmonic measure and panharmonic measure enable accurate simulation in mixed potential domains?
• RQ5 What is the numerical accuracy and convergence behavior of the KWOS algorithm in one- and two-dimensional test cases?

Key findings

• The KWOS algorithm provides a robust and efficient method for simulating weak solutions of the Schrödinger equation with piecewise constant positive potentials.
• The method achieves high accuracy in one-dimensional examples, with the approximate solution $ \hat{u}_{1000} $ closely matching the analytical solution in Example 4.3.
• In Example 4.4, the KWOS algorithm successfully approximated the solution of the Yukawa equation on a non-convex domain with $ K=1000 $ particles.
• For the mixed Laplace-Yukawa problem in Example 4.5, the algorithm produced a stable and accurate approximation $ \hat{u}_{250} $, demonstrating robustness in complex geometries.
• The use of panharmonic measures and Radon-Nikodym derivatives enables exact representation of solutions via stochastic expectations, avoiding numerical differentiation.
• The algorithm avoids direct exit time estimation by interpreting potential as killing, significantly improving computational efficiency over traditional methods.