Quasi-Monte Carlo time-splitting methods for the Schr\"odinger equation with Gaussian random potential

In this paper, we study the Schrödinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables.

The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis.

Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space.

Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling.

This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP.

We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples.

Numerical experiments illustrate the sharpness of the error estimate.