Derivation of the 1d Gross–Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons
Boßmann L, Teufel S (2019)
Publication Type: Journal article
Publication year: 2019
Journal
Book Volume: 20
Pages Range: 1003-1049
Journal Issue: 3
DOI: 10.1007/s00023-018-0738-7
Abstract
We consider the dynamics of N interacting bosons initially forming a Bose–Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of order ε. The non-negative interaction potential is scaled such that its range and its scattering length are both of order (N/ε2)-1, corresponding to the Gross–Pitaevskii scaling of a dilute Bose gas. We show that in the simultaneous limit N→ ∞ and ε→ 0 , the dynamics preserve condensation and the time evolution is asymptotically described by a Gross–Pitaevskii equation in one dimension. The strength of the nonlinearity is given by the scattering length of the unscaled interaction, multiplied with a factor depending on the shape of the confining potential. For our analysis, we adapt a method by Pickl (Rev Math Phys 27(01):1550003, 2015) to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour in Boßmann (Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons. arXiv preprint, 2018. arXiv:1803.11011).
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APA:
Boßmann, L., & Teufel, S. (2019). Derivation of the 1d Gross–Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons. Annales Henri Poincaré, 20(3), 1003-1049. https://doi.org/10.1007/s00023-018-0738-7
MLA:
Boßmann, Lea, and Stefan Teufel. "Derivation of the 1d Gross–Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons." Annales Henri Poincaré 20.3 (2019): 1003-1049.
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