Landau – Zener problem for energies close to potential crossing points

We examine one overlooked in previous investigations aspect of well-known Landau – Zener (LZ) problem, namely, the behavior in the intermediate, i.e. close to a crossing point, energy region, when all four LZ states are coupled and should be taken into account. We calculate the $4 \times 4$ connection matrix in this intermediate energy region, possessing the same block structure as the known connection matrices for the tunneling and in the over-barrier regions of the energy, and continously matching those in the corresponding energy regions. Applications of the results may concern the various systems of physics, chemistry or biology, ranging from molecular magnets and glasses to Bose condensed atomic gases.