Stokes constant

In terms of the Stokes constant U, the reduced scattering matrix can be quantum mechanically exactly given by... 

The Stokes constant U, which is actually a function of the parameters, is given as... 

The reduced scattering matrix in terms of the Stokes constant U is given quantum mechanically exactly as... 

The semiclassical expressions in the ZN theory are given below for the Stokes constant and other imporatant physical quantities. 

B.6 CYLINDRICAL NAVIER-STOKES, CONSTANT VISCOSITY B.6.1 -Momentum, Constant Viscosity... 

The very basic mathematics, i.e., Stokes phenomenon, which underlies semiclassical theory, is briefly explained in this section by taking the Airy function as an example. The Stokes constant and connection matrix in the case of the Weber function are provided, since the Weber function is useful in many applications. Finally, the Stokes phenomenon of the linear curve-crossing model discussed in Sec. IV is explained briefly. 

The ordinary Airy function A,(z) corresponds to this solution with A = 0. Equation (85) represents the famous connection formula for the WKB solutions crossing the turning point. As can now be easily understood, once we know all the Stokes constants the connections among asymptotic solutions are known and the physical quantities, such as the scattering matrix, can be derived. However, the Airy function is exceptionally simple and the Stokes constants are generally not known except for some special cases (40). 

When the coefficient of the differential equation is an nth-order polynomial, the n + 2 Stokes lines run radially in the asymptotic region. There are thus n + 2 unknown Stokes constants, but only three independent conditions are obtained from the singlevaluedness as demonstrated for the Airy function. 

In this section the mathematical procedure is briefly outlined for deriving the exact expressions of the reduced scattering matrices and of the Stokes constants in the linear curve-crossing problems and for devising new semiclassical approximations to them. 

Using the transformation in Eq. (106), we can obtain the simple relations between the Stokes constant Uj in the 2-plane and the Stokes constant 7 in the -plane for j = 1-6. Using the symmetry of the differential equation (107) with Eq. (108), we can derive the interrelations among T/s as... 

The above connection is made on the Stokes line and one-half of the Stokes constants are assigned to the Stokes line according to the Jeffrey s connection formula (24). This procedure leads to the explicit approximate expression of Uu... 

In the following three subsections the Stokes constant U and other basic quantities are presented. Finally in Sec. VI.A.4, the interesting phenomenon of complete reflection is further interpreted. 

C. Zhu and H. Nakamura, The two-state linear curve crossing problems revisited. II Analytical approximations for the Stokes constant and scattering matrix The Landau-Zener case, J. Chem. Phys. 97 8497 (1992). 

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