Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons
Boßmann L (2020)
Publication Type: Journal article
Publication year: 2020
Journal
Book Volume: 238
Pages Range: 541-606
Journal Issue: 2
DOI: 10.1007/s00205-020-01548-w
Abstract
We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε. The interaction is non-negative and scaled in such a way that its scattering length is of order ε/ N, while its range is proportional to (ε/ N) β with scaling parameter β∈ (0 , 1]. We consider the simultaneous limit (N, ε) → (∞, 0) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β. For β∈ (0 , 1) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β= 1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.
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APA:
Boßmann, L. (2020). Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons. Archive for Rational Mechanics and Analysis, 238(2), 541-606. https://doi.org/10.1007/s00205-020-01548-w
MLA:
Boßmann, Lea. "Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons." Archive for Rational Mechanics and Analysis 238.2 (2020): 541-606.
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