  The doping dependent isotope effect on the critical temperature (Tc) is
calculated for multi-band multi-condensate superconductivity near a 2.5
Lifshitz transition. We focus on multi-band effects that arises in
nano-structures and in density wave metals (like spin density wave or charge
density wave) as a result of the band folding. We consider a superlattice of
quantum stripes with finite hopping between stripes near a 2.5 Lifshitz
transition for appearing of a new sub-band making a circular electron-like
Fermi surface pocket. We describe a particular type of BEC (Bose-Einstein
Condensate) to BCS (Bardeen-Cooper-Schrieffer condensate) crossover in
multi-band / multi-condensate superconductivity at a metal-to-metal transition
that is quite different from the standard BEC-BCS crossover at an
insulator-to-metal transition. The electron wave-functions are obtained by
solving the Schr\"odinger equation for a one-dimensional modulated potential
barrier. The k-dependent and energy dependent superconducting gaps are
calculated using the k-dependent anisotropic Bardeen-Cooper-Schrieffer (BCS)
multi-gap equations solved joint with the density equation, according with the
Leggett approach currently used now in ultracold fermionic gases. The results
show that the isotope coefficient strongly deviates from the standard BCS value
0.5, when the chemical potential is tuned at the 2.5 Lifshitz transition for
the metal-to-metal transition. The critical temperature Tc shows a minimum due
to the Fano antiresonance in the superconducting gaps and the isotope
coefficient diverges at the point where a BEC coexists with a BCS condensate.
On the contrary Tc reaches its maximum and the isotope coefficient vanishes at
the crossover from a polaronic condensate to a BCS condensate in the new
appearing sub-band.
