# Derivation of the time dependent Gross–Pitaevskii equation in two dimensions

Jeblick M, Leopold NK, Pickl P. 2019. Derivation of the time dependent Gross–Pitaevskii equation in two dimensions. Communications in Mathematical Physics. 372(1), 1–69.

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Author

Jeblick, Maximilian;
Leopold, Nikolai K

^{ISTA}^{}; Pickl, PeterDepartment

Abstract

We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting froman interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx), for any β>0, or to be given by VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

Publishing Year

Date Published

2019-11-08

Journal Title

Communications in Mathematical Physics

Acknowledgement

OA fund by IST Austria

Volume

372

Issue

1

Page

1-69

ISSN

eISSN

IST-REx-ID

### Cite this

Jeblick M, Leopold NK, Pickl P. Derivation of the time dependent Gross–Pitaevskii equation in two dimensions.

*Communications in Mathematical Physics*. 2019;372(1):1-69. doi:10.1007/s00220-019-03599-xJeblick, M., Leopold, N. K., & Pickl, P. (2019). Derivation of the time dependent Gross–Pitaevskii equation in two dimensions.

*Communications in Mathematical Physics*. Springer Nature. https://doi.org/10.1007/s00220-019-03599-xJeblick, Maximilian, Nikolai K Leopold, and Peter Pickl. “Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions.”

*Communications in Mathematical Physics*. Springer Nature, 2019. https://doi.org/10.1007/s00220-019-03599-x.M. Jeblick, N. K. Leopold, and P. Pickl, “Derivation of the time dependent Gross–Pitaevskii equation in two dimensions,”

*Communications in Mathematical Physics*, vol. 372, no. 1. Springer Nature, pp. 1–69, 2019.Jeblick, Maximilian, et al. “Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions.”

*Communications in Mathematical Physics*, vol. 372, no. 1, Springer Nature, 2019, pp. 1–69, doi:10.1007/s00220-019-03599-x.**All files available under the following license(s):**

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2019-11-25

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