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arXiv · 1903.08740

Gaussian wave packet transform based numerical scheme for the semi-classical Schrödinger equation with random inputs

Shi Jin·Liu Liu·Giovanni Russo·Zhennan Zhou
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Abstract

In this work, we study the semi-classical limit of the Schrödinger equation with random inputs, and show that the semi-classical Schrödinger equation produces $O(\varepsilon)$ oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ODE system for the wave packet parameters coupled with a PDE for the quantity $w$ in rescaled variables. Further, we show that the $w$ equation does not produce $\varepsilon$ dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, i.e. simulating the $w$ equation, it is sufficient to use $\varepsilon$ independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions.

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Shi Jin, Liu Liu, Giovanni Russo, Zhennan Zhou. 2019-10-10. Gaussian wave packet transform based numerical scheme for the semi-classical Schrödinger equation with random inputs. https://arxiv.org/abs/1903.08740

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