Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings

Clotilde Fermanian Kammerer; Caroline Lasser; Didier Robert

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arXiv · 2305.17053

Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings

Clotilde Fermanian Kammerer·Caroline Lasser·Didier Robert

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Abstract

This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schr\"odinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schr\"odinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter.

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BibTeX RIS

Clotilde Fermanian Kammerer, Caroline Lasser, Didier Robert. 2023-05-26. Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings. https://arxiv.org/abs/2305.17053

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