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g$_{∗}$ is approached. When g = g − ∗ , i.e. just less than g$_{∗}$ , then a → ∞ , whereas when g = g + ∗ , a → −∞ . The case g$_{R}$ = −∞ is on the BCS side of the crossover, whereas g$_{R}$ = + ∞ is on the BEC side. See Figure 1. 

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The S-matrix (4) has a pole at k = 16 πi/mg$_{R}$ . Since physical bound states correspond to poles at Im( k ) > 0, the bound state exists only on the BEC side of the critical point, with energy 

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In the BEC to BCS crossover from 1 /a = 0 + to 1 /a = 1 / 0 − the binding energy goes to zero and the bound state disappears. Nevertheless, the crossover is expected to be smooth, if on the BEC side one includes the bound state in the thermodynamics. 

