The two-state linear curve crossing problems revisited. I. Analysis of Stokes phenomenon and expressions for scattering matrices

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Abstract
The classic problems of the two-state linear curve crossing both for the same and the opposite sign of slopes of the linear diabatic potentials are analyzed in a unified way by exactly dealing with the Stokes phenomenon for the four transition points of the associated second-order differential equations. First, distributions of the transition points and the Stokes lines are fully clarified for the whole range of the two parameters which effectively represent the coupling strength and the collision energy. Secondly, the so-called reduced scattering matrices are found to be expressed in terms of only one (complex) Stokes constant. This is made possible by finding the relations among the six Stokes constants. Finally, this one Stokes constant is given exactly and analytically by a convergent infinite series.