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Threshold expansion formula of $N$-boson in finite volume from 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 sub-leading non-zero momenta mode contributions are connected through finite volume Faddeev-type coupled dynamical equations. Eliminating non-zero momenta modes by iterations ultimately yields an analytic expression that can be solved by threshold expansion.