Dynamical low-rank approximation for the semiclassical Schrodinger equation with uncertainties

TL;DR

This paper introduces a dynamical low-rank approximation method for efficiently solving the semiclassical Schrödinger equation with uncertainties, capturing quantum dynamics with fewer basis functions despite complex oscillations.

Contribution

The paper extends two robust integrators to the semiclassical regime and demonstrates their efficiency and accuracy in capturing low-rank structures in uncertain quantum systems.

Findings

DLR method is more efficient than stochastic Galerkin methods.

Wave functions remain in low-rank subspaces despite oscillations.

High fidelity achieved with small, robust numerical ranks.

Abstract

In this paper, we propose a dynamical low-rank (DLR) approximation framework for solving the semiclassical Schrodinger equation with uncertainties. The primary numerical challenges arise from the dual nature of the oscillations: the spatial oscillations inherent in the semiclassical limit and the high-frequency oscillations in the random space induced by uncertainties. We extend two robust integrators -- the projector-splitting integrator and the unconventional integrator -- to the semiclassical regime to evolve the solution on a low-rank manifold. Through extensive numerical experiments, we demonstrate that the DLR method is significantly more computationally efficient than the standard stochastic Galerkin method, as it captures the essential quantum dynamics using a much smaller number of basis functions. Our findings reveal that despite the complex oscillatory patterns of the wave function, its evolution remains concentrated in a low-rank subspace for the cases investigated. Specifically, we observe that the DLR method achieves high fidelity with a remarkably small numerical rank, which remains robust even as the semiclassical parameter $\varepsilon$ decreases. Within our problem settings, the results further suggest that the rank growth is primarily driven by the randomness and regularity of the potential. These results provide practical insights into the low-rank structure of uncertain quantum systems and offer an efficient approach for high-dimensional uncertainty quantification in the semiclassical regime.