Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons

Abstract

AbstractWe 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 $$\varepsilon $$ ε . The interaction is non-negative and scaled in such a way that its scattering length is of order $$\varepsilon /N$$ ε / N , while its range is proportional to $$(\varepsilon /N)^{\beta }$$ ( ε / N ) β with scaling parameter $$\beta \in (0,1]$$ β ∈ ( 0 , 1 ] . We consider the simultaneous limit $$(N,\varepsilon )\rightarrow (\infty ,0)$$ ( 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 $$\beta $$ β . For $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice $$\beta =1$$ 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.