If only two-body interaction is considered then what's the energy if I put three particles on one site?
Assume delta-interaction and interaction strength is proportional to scattering length $a_F$ which depends on the total spin of the two-particle state.
For two particle case, the interaction is simply $\hat U |F,M\rangle = U_F |F,M\rangle $
Suppose the three particles are fermions, then I will take a completely anti-symmetrized spin wave function. But now, how to express the interaction energy in terms of two-body scattering lengths?