ergy is same as in vacuum, E$_{b}$ ( T$_{p}$ ) = E$_{b}$$_{0}$ = ¯ h $^{2}$/ ( ma 2 $_{s}$). At finite temperatures, molecules not only exist for a$_{s}$ > 0, but also appear as loosely- bound pairs for a$_{s}$ < 0 in the mean-field theory, whereas in vacuum Feshbach molecules exist only for a$_{s}$ > 0. However at zero temperature, the molecular-BEC state exist only when 1 / ( n 1 / $^{3}$a$_{s}$ ) > 3 . 94 [12]. In Fig. 1, the molecular binding energy in the mean-field approximation is plotted at both T = T$_{p}$ and T = 0, in comparison with its vacuum value. The temperature dependence of the binding energy for different scattering lengths is shown in the inset. For 1 / ( n 1 / $^{3}$a$_{s}$ ) < 3 . 94, the binding energy decreases monotonically with temperature, all the way to zero at the lower boundary of the molecular-BEC phase; for 1 / ( n 1 / $^{3}$a$_{s}$ ) > 3 . 94, the binding energy decreases monotonically to a finite value at zero temperature. In the dilute limit, n 1 / $^{3}$a$_{s}$ → 0 $^{+}$, the temperature dependence of the binding energy is weak, and the binding energy is approximately given by vacuum value E$_{b}$$_{0}$ . A mechanically stable Bose gas has positive compressibility, indicated by ∂µ/∂n > 0. In the normal state, a Bose gas with a$_{s}$ < 0 collapses when ∂µ/∂n = 0. In the Hartree-Fock approximation, this condition can be rewritten as The collapse temperature T$_{C}$ is always higher than the molecular BEC transition temper- ature T$_{p}$ [15–17]. In the limit of weakly-attractive interaction, n 1 / $^{3}$a$_{s}$ → 0 $^{−}$, the collapse temperature T$_{C}$ approaches to the ideal atomic BEC temperature T$_{a}$ , In contrast, a normal Bose gas with a$_{s}$ > 0 is always mechanically stable. We examine the compressibility of the molecular-BEC state numerically following Eq. (4) and (5). In Fig. 2, the chemical potential µ is plotted as a function of density n at given temperatures and scattering lengths. For a$_{s}$ < 0, the chemical potential µ is monotonically decreasing with the increase of n , ∂µ/∂n < 0, showing that the system is mechanically unstable as found in previous studies [15, 16]. For a$_{s}$ > 0, the compressibility is always positive, ∂µ/∂n > 0, and the molecular-BEC state is mechanically stable. Therefore, in a − k$_{B}$T$_{C}$ g = 1 2 ∫ d $^{3}$k (2 π ) 3 csch 2 ( ϵ$_{k}$ − µ ′ 2 k$_{B}$T$_{C}$ ) . (10) T$_{C}$ = T$_{a}$ [ 1 + 16 π 3 ζ 4 / $^{3}$( 3 $_{2}$) n 1 / $^{3}$| a$_{s}$ | ] . (11) D. Mechanical stability 6