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Pré-Publication, Document De Travail Année : 2019

Derivation of the time-dependent Gross-Pitaevskii equation for the dipolar gases

Résumé

We derive the time-dependent dipolar Gross-Pitaevskii (GP) equation from the N-body Schrödinger equation. More precisely we show a norm approximation for the solution of the many body equation as well as the convergence of its one-body reduced density matrix towards the orthogonal projector onto the solution of the dipolar GP equation. We consider the interpolation regime where interaction potential is scaled like $N^{3\beta−1} w(N^\beta (x − y))$, the range of validity of $\beta$ depends on the stability of the ground state problem. In particular we can prove the convergence on the one-body density matrix assuming $\widehat{w} ≥ 0$ and $\beta < 3/8$.
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hal-02091613 , version 1 (05-04-2019)

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Arnaud Triay. Derivation of the time-dependent Gross-Pitaevskii equation for the dipolar gases. 2019. ⟨hal-02091613⟩
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