as the Thouless criterion which marks the pairing instability of the normal phase, The density equation (5) at T$_{p}$ becomes the Hartree-Fock self-consistency condition of the normal Bose gas In the limit of the weakly-attractive interaction, n 1 / $^{3}$a$_{s}$ → 0 $^{−}$, the transition temperature T$_{p}$ is close to the ideal atomic BEC temperature T$_{a}$ = (2 π ¯ h $^{2}$/mk$_{B}$ )[ n/ζ (3 / 2)] 2 / $^{3}$, as found in a previous study [17]. In the opposite limit, n 1 / $^{3}$a$_{s}$ → 0 $^{+}$, molecules are tightly bound with the binding energy approximately given by E$_{b}$$_{0}$ . From Eq. (12), the chemical potential is approximately given by µ = − E$_{b}$$_{0}$/ 2. The solution of density equation (13) yields which is essentially the same as the mean-field result about the BEC limit of a Fermi gas with a BEC-BCS crossover. As pointed out in earlier works [19], the divergent mean-field transition temperature in this limit in fact corresponds to the temperature of molecule dis- sociation, not molecular condensation. The transition temperature T$_{p}$ can be renormalized by pairing fluctuations [20]. The lower boundary of the molecular-BEC phase is determined by Δ = − µ $^{′}$, where the excitation energy becomes gapless, E$_{k}$ = √ ϵ$_{k}$ ( ϵ$_{k}$ − 2 µ $^{′}$). At zero temperature, this boundary is located at 1 / ( n 1 / $^{3}$a$_{s}$ ) = 4(3 /π ) 1 / 3 ≃ 3 . 94 [12]. Disappearance of gap in excitation energies was interpreted as appearance of atomic condensation [15, 20]. However, as discussed later in section III, a coherent mixture with atoms and molecules is mechanically unstable. The order parameter Δ in the molecular-BEC phase evolves smoothly between two phase boundaries. Based on the above results, the mean-field phase diagram is plotted in Fig. 3. The molecular-BEC phase is labeled as P(S) for a$_{s}$ > 0 and P(C) for a$_{s}$ < 0, where symbols (S) and (C) represent mechanical stability and mechanical collapse. 8 k$_{B}$T$_{p}$ = E$_{b}$$_{0}$ ln − $^{1}$( E$_{b}$$_{0}$/k$_{B}$T$_{a}$ ) / 3 , (15) T$_{p}$ = T$_{a}$ [ 1 + 8 π 3 ζ 4 / $^{3}$( 3 $_{2}$) n 1 / $^{3}$| a$_{s}$ | ] , (14) n = ∫ d $^{3}$k (2 π ) 3 1 e ( ϵ$_{k}$ − µ ′ ) /k B T$_{p}$ − 1 . (13) − 1 g = ∫ d $^{3}$k (2 π ) 3 [ 1 2( ϵ$_{k}$ − µ $^{′}$) coth ( ϵ$_{k}$ − µ ′ 2 k$_{B}$T$_{p}$ ) − 1 2 ϵ$_{k}$ ] . (12)