Derivation of the 2d GrossPitaevskii Equation for Strongly Confined 3d Bosons
Abstract
We study the dynamics of a system of N interacting bosons in a discshaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ∊ . The interaction is nonnegative and scaled in such a way that its scattering length is of order ∊ /N , while its range is proportional to (∊^{/N ) β} with scaling parameter β ∈(0 ,1 ] . We consider the simultaneous limit (N ,∊ )→(∞ ,0 ) and assume that the system initially exhibits BoseEinstein condensation. We prove that condensation is preserved by the Nbody dynamics, where the timeevolved condensate wave function is the solution of a twodimensional nonlinear equation. The strength of the nonlinearity depends on the scaling parameter β . For β ∈(0 ,1 ) , we obtain a cubic defocusing nonlinear Schrödinger equation, while the choice β =1 yields a GrossPitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.
 Publication:

Archive for Rational Mechanics and Analysis
 Pub Date:
 July 2020
 DOI:
 10.1007/s0020502001548w
 arXiv:
 arXiv:1907.04547
 Bibcode:
 2020ArRMA.238..541B
 Keywords:

 Mathematical Physics
 EPrint:
 Arch. Ration. Mech. Anal. 238(2) (2020)