Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons - Boßmann, Lea (Archive for Rational Mechanics and Analysis, 2020)
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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