Multilevel Landau–Zener formulae: adiabatic reduction on a complex path

We consider, in the semi-classical (adiabatic) limit, evolution equations whose generators extend into a strip around real axis as a holomorphic family of operators (with respect to the time-variable). The asymptotic expansion of the $\mathbb S$-matrix associated to this evolution can be expressed in terms of simple quantities attached to the singularities for the spectrum of Hamiltonians from complex-time plane. We extend to many-level case the result from [26] which contains as limit cases both the Landau–Zener formula and Friedrichs–Hagedorn results for this problem.