Threshold expansion formula of N bosons in a finite volume from a variational approach

In present work, we show how the threshold expansion formula of N identical bosons in finite volume may be derived by iterations of Faddeev-type coupled dynamical equations. The energy shift of N-boson system near threshold is dominated by zero momenta mode of N-body amplitudes with all particles nearly static. The dominant zero momenta mode and subleading nonzero momenta mode contributions are connected through finite volume Faddeev-type coupled dynamical equations. Eliminating nonzero momenta modes by iterations ultimately yields an analytic expression that can be solved by threshold expansion.