  We investigate by means of Dynamical Mean-Field Theory the crossover from BCS
superconductivity to Bose-Einstein (BE) condensation of preformed pairs in the
attractive Hubbard model. We follow the evolution of the two energy scales
underlying the superconducting phenomenon, the gap $\Delta_0$ and the
superfluid stiffness $D_S$, which controls the phase coherence. The BCS-BE
crossover is clearly mirrored in a change in the hierarchy of these two scales,
the smallest of the two controlling the critical temperature. In the whole
intermediate-to-strong coupling region $T_c$ scales with $D_S$, while $T_C$ is
proportional to $\Delta_0$ only in the BCS regime. This evolution as a function
of the interaction qualitatively resembles what happens in the cuprates when
the doping is decreased towards the Mott insulator.
  This continuous change reflects also in the energetic balance at the
superconducting transition. While superconductivity is stabilized by a
potential energy gain in the BCS regime, the strong-coupling superconductivity
is made stable by a reduction of kinetic energy. Interestingly the
intermediate-coupling region, where the maximum $T_c$ is achieved, behaves
similarly to the strong-coupling regime. The above finding implies that the
attractive Hubbard model can account qualitatively for the anomalous behavior
of optical spectra around $T_c$, where an increase of spectral weight is
observed in under and optimally doped cuprates, while the overdoped samples
have a more standard behavior. This qualitative agreement is lost in the normal
phase, specifically at strong-coupling, calling for the inclusion of strong
correlation effects in the theoretical description.
