on_{\star}=-2/mr_{\star}^2$. After resonance when $\delta' \to +\infty$, the CI bound state energy behaves as $\epsilon_b'\simeq -1/ma^2$, which translates into $\epsilon_b'/\epsilon_{\star}'\simeq (\delta'+A)^2/2\simeq
\delta^{'2}/2$ in terms of the parameter $\delta'$ introduced above. Similarly, in the BFRM, the bound state energy decreases monotonically with the behavior $\epsilon_b/\epsilon_{\star} \simeq
\sqrt{2}\delta$ as a function of the dimensionless detuning $\delta$.

![Confinement induced bound state energy $\epsilon_b'$ \[in units of the bound state energy on resonance $|\epsilon_{\star}'|$\] as a function of the dimensionless parameter $\delta'$ (full line). The dashed line corresponds to the asymptotic behavior $\epsilon_b' \simeq -mg_1^{'2}/4$ and the dotted line to the asymptotic behavior $\epsilon_b' \simeq -1/ma^2$.](BMOboundstate.eps "fig:"){height="6cm"} \[BMOboundstate\]

Many-body problem
=================

The grand partition function at temperature $T\equiv 1/\beta$ and chemical potential $\mu$ can be written as a path integral $$\begin{aligned}
Z=\int \mathcal{D}(\bar{\psi}_{\sigma},\psi_{\sigma})
\mathcal{D}(\bar{\psi}_B,\psi_B) e^{-S}
\label{Z}\end{aligned}$$ over Grassmann fields $\bar{\psi}_{\sigma}$, $\psi_{\sigma}$ with $\sigma=\uparrow,\downarrow$ and complex fields $\bar{\psi}_B$, $\psi_B$ [@AS]. The action corresponding to the Hamiltonian (\[BFM\]) is: $$\begin{aligned}
&S&=\int_0^{\beta}d\tau \int dx \bigg(
\sum_{\sigma={\uparrow,\downarrow}} \bar{\psi}_{\sigma}
\Big[\partial_{\tau}-\frac{\partial_x^2}{2m}-\mu\Big]\psi_{\sigma}
\nonumber \\
&+&\bar{\psi}_{B}\Big[\partial_{\tau}-\frac{\partial_x^2}{4m}-2\mu+\nu\Big]
\psi_{B} + g \Big( \bar{\psi}_{B}\psi_{\uparrow}
\psi_{\downarrow}+c.c.\Big)\bigg).\nonumber\\ \left. \right.
\label{originalaction}\end{aligned}$$ The average total number of atoms is obtained from $$\begin{aligned}
\langle N \rangle = \frac{\partial F}{\partial \mu}\end{aligned}$$ where $F\equiv-T \ln Z$ is the grand potential and we are interested in the $T\to 0$