 Effective two-dimensional contact coupling

Factorizing a normalized condensate field as ${\Psi }_{3D}(x,y,z)=\psi (x,y)\chi (z)$ with $\int |\chi {|}^{2}dz=1$ reduces its contact interaction energy to $(g_{2D}/2)\int |\psi {|}^{4}dxdy$, with the displayed coupling. Thus $g_{2D}$ has units of energy times area, whereas $g_{3D}=4\pi {\hslash }^{2}a/m$ has units of energy times volume for a physical scattering length $a$. For a normalized axial Gaussian $\chi (z)={\pi }^{-1/4}{a}_z^{-1/2}{e}^{-z^2/(2a_z^2)}$, direct Gaussian integration gives $g_{2D}=g_{3D}/(\sqrt{2\pi }{a}_z)$. A model with density normalized per unit axial length can instead retain $g_{3D}$ with a line-density normalization.