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arXiv · 1905.06904

Rank-1 lattices and higher-order exponential splitting for the time-dependent Schr\"odinger equation

Yuya Suzuki·Dirk Nuyens
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Abstract

In this paper, we propose a numerical method to approximate the solution of the time-dependent Schr\"odinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral method on rank-$1$ lattice points, and then discretize time by using a higher-order exponential operator splitting method. In this scheme the convergence rate of the time discretization depends on properties of the spatial discretization. We prove that the proposed method, using rank-$1$ lattice points in space, allows to obtain higher-order time convergence, and, additionally, that the necessary condition on the space discretization can be independent of the problem dimension $d$. We illustrate our method by numerical results from 2 to 8 dimensions which show that such higher-order convergence can really be obtained in practice.

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Yuya Suzuki, Dirk Nuyens. 2019-05-16. Rank-1 lattices and higher-order exponential splitting for the time-dependent Schr\"odinger equation. https://arxiv.org/abs/1905.06904

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