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We discuss the shape resonance in the superconducting gaps of a two bandsuperconductor by tuning the chemical potential at a Lifshitz transition forFemi surface neck collapsing and for spot appearing. The high temperaturesuperconducting scenario for complex matter shows the coexistence of a firstBCS condensate made of Cooper pairs in the first band and a second boson-likecondensate made of bosons like bipolarons, in the second band where thechemical potential is tuned near a Lifshitz transition. The interband couplingcontrols the shape resonance in the pair exchange between the two condensates.We discuss the particular BCS-Bose crossover that occurs at the shape resonancetuning the Lifshitz parameter (the energy difference between the chemicalpotential and the Lifshitz topological transition) like tuning the externalmagnetic field for the Feshbach resonances in ultracold gases. Thissuperconducting phase provides a particular case of topologicalsuperconductivity with multiple condensates of different winding numbers

Shape resonance at a Lifshitz transition for high temperature superconductivity in multiband superconductors