The Stokes phenomena of the standard second-order ordinary differential equations with the coefficient functions of the certain n-th order polynomials are investigated. Four cases of the coefficient function q(z) are solved to find analytical solutions of the Stokes constants: (i) q(z) = a2Nz2N+ΣN-1 j=-∞ ajzj;(ii) q(z) =a2N-1z2N-1+ΣN-2 j=-∞ ajzi; (ii) q(z) = a2N-1 z 2N-1 +ΣN-2j=-∞ ajzj; (iii) q(z) = Σ4j=0ajzj; and (iv) q(z) =Σ3j=0 ajzj. The case (iii) can be immediately applied to the two-state linear curve crossing problems which represent the most basic models for non-adiabatic transition processes in atomic and molecular physics. The two-state linear curve crossing problems are generally classified into the following two cases: (1) the same sign of slopes of two diabatic potential curves(Landau-Zener case), and (2) the opposite sign of slopes(nonadiabatic tunneling case). The reduced scattering matrix for each case has been found to be expressed in terms of only one Stokes constant U1 which is solved exactly and analytically in a form of convergent infinite series. This means that exact quantal solutions of the reduced scattering matrices for both cases are analytically found for the first time. Furthermore, new semiclassical solutions of the reduced scattering matrices for both cases are derived in simple compact forms. Especially, the case that the collision energy is lower than the crossing point is colrectly dealt with fol the first time. Both quantal and semiclassical solutions for the reduced scattering matrix are made possible by expressing the connection matrix, which is a crucial bridge to link physics and mathematics, in terms of Stokes constants. Among the fruitful results obtained, one of the most notable ones is about a derivation of a new formula to replace the widely used Landau-Zener formula for nonadiabatic transition probability. The new one is as simple as the Landau-Zener, but works much better than the latter. On the the other hand, by fully analyzing the distributions of the four tlansition points and the Stokes lines in complex plane for the basic equations of the two-state linear curve crossing problems, the validity conditions are made clear for the present and the other available semiclassical formulas of the reduced scattering matrices.
Chapter 1
This thesis begins with the asymptotic solutions of the second-order differential equations for the four cases mentioned above. The asymptotic solutions are found exactly in the form of infinite series, in which the recurrence relations among the coefficlents are given explicitly. This is made possible by transforming the original differential equations from the complex-z plane to a new complex-ε plane in which all the Stokes lines coincide with the real axis. At the same time the standard asymptotic WKB solutions are introduced fol convenience as reference functions to define Stokes constants. The Stokes phenomenon is reviewed and explained briefly so that physicists and chemists can get quickly an insight on the topics discussed in this thesis.
Chapter 2
A central task in the subject of Stokes phenomenon is to find analytical solutions of Stokes constants. The standard asymptotic WKB solutions are proved to be quite useful for the present type of analysis, especially for deriving the relations among Stokes constants. Actually, three inde-pendent relations for all Stokes constants U defined in the complex-z plane are easily established. They are very useful for many physical problems although they are not enough to have a complete. A further deduction is made by transforming the asymptotic solutions from complex-z plane to the complex-ε plane where the Stokes constants Ti are defined. One-to-one simple correspondence is obtained between Ui and Ti. What is fascinating about the complex-ε plane is that all Stokes constants Ti can be simply related to only one, for instance, T1､by using a particular transformation under which the differential equation in the complex-ε plane is invariant. The conclusions obtained up to now hold not only for the four cases mentioned above but generally. The remaining most difficult problem is how to find an analytical solution for T1 for each case. By generalizing the coupled-wave-integral- equations method devised by Hinton, Stokes constant T1 is finally shown to be expressed in the analytical form of a convergent infinite series as a function of the coefficients q(z) for all foul cases.
Chapter 3
A connection matrix presented in this chapter represents an important physical quantity i.e., scattering matrix, and bridge between the Stokes phenomenon in mathematics and the two-state linear curve crossing problems in physics. If the standard WKB solutions are used in the asymptotic legion ｜z｜ →∞ of the complex plane, the connection matrix is exactly expressed in terms of the Stokes constants. This matrix can connect solutions in one asymptotic region in complex plane to solutions in another asymptotic region, such as physical important connections between two anti-Stokes lines, two Stokes lines, and one anti-Stokes and Stokes lines. What is fascinating about expressing the connection matrix in terms of Stokes constants is as follows: A physically required connection matrix sometimes can not be well-approximated by following traditional semiclassical path. It is much more flexible and versatile to try to find Stokes constants. Based on the knowledge of the distributions of transition points and Stokes lines, such a path which may not correspond to the physical connection matrix can be designed to derive the best semiclassical solution from Stokes constants. Excellent examples will be given in chapters 5 , 6 and 7 for semiclassical solutions of the reduced scattering matrices for the cases of energy lower than the crossing points. The connection problems for one transition point and two tlansition points are briefly reviewed, and those for three transition points and four transition points are presented in detail. The last case is mainly concerned with the curve crossing problems discussed in the subsequent chapters.
Chapter 4
The classic problems of the two-state linear curve crossing were initially discussed by Lan-dau, Zener and Stueckelberg. As mentioned before, there are the following two cases: (1) the same sign of slopes of two diabatic potentials(Landau- Zener case), and (2) the opposite sign of slopes(nonadiabatic tunneling case). It is well known that the reduced scattering matrices for these two problems can be described in terms of the two parameters a2(effective coupling strength) and b2(effective collision energy). Finding the exact analytical quantal solutions for the reduced scat-tering matrices is very challenging and very difiicult question. The answer to this question is given in this chapter. The starting point is the basic differential equation of the case (iii) mentioned before. By using the connection matrix obtained in chapter 3, the reduced scattering matrix for each case is first expressed in terms of three Stokes constants. Then by taking into account two extra conditions in addition to the unitarity of reduced scattering matrix, it is shown to be expressed finally in terms of only one Stokes constant U1. Finally, this one Stokes constant is given exactly and analytically by a convergent infinite series which is a direct result from chapter 2. Another work reported in this chapter is a new numerical method to solve reduced scattering matrix for the nonadiabatic tunneling case. The original coupled equations suffer from very rapid oscillation asymptotically and can not give stable and reliable numerical results. New coupled equations are presented which involve ordinary sine and cosine solutions asymptotically. Numerical results of reduced scattering matrix can be obtained with any desirable accuracy.
Chapter 5
The distributious of the four transition points and the Stokes lines are fully analyzed for both Landau-Zener and nonadiabatic tunneling cases in the whole plane of the two parameters a2 and b2. This analysis is, of course, important in itself, but what is Inore significant about this is that the structure of the distributions essentially determines which path in complex plane is the best for obtaining good semiclassical solutions of the reduced scattering matrices. The semiclassical method used here and in the following chapters should be potentially useful for other problems in physics and chemistry.
Chapter 6
The semiclassical solution of the reduced scattering matrix for the Landau-Zener case is obtained in this chapter. Since the reduced scattering matrix is expressed in terms of one Stokes constant U1 in chapter 4, question now is how to find an approximate solution for U1. The distributions of transition points and Stokes lines analyzed in chapter 5 clearly show that there are two best choices of path to get good approxirnate solutions of U1. One path corresponds to the connection on the anti-Stokes lines along which the four transition points are separated in two paris. Another path corresponds to the connection on the Stokes lines along which the foul transition points are again separated in two pairs. The former(latter) corresponds to high(low) energy limit. In each limiting case, the exact connection matrix can be approximately decomposed into a product of the two matrices, each of which represents the connection matrix based on two transition points as is given in chapter 3. Finally, two new compact analytical formulas for the reduced scattering matrix are derived and compared with available ones. The a2 - b2 plane is now divided into five regions, in each one of which the best recommended formulas are proposed. The new formulas proposed are simple and explicit functions of the two parameters a2 and b2. Especially, a simple formula which works much better than the conventional Landau-Zener formula is obtained for nonadiabatic transition probability for one passage of crossing point.
Chapter 7
The semiclassical solution of the reduced scattering matrix for the nonadiabatic tunneling case is obtained in this chapter. The reduced scattering matrix is, of course, given in chapter 4 in terms of one Stokes constant U1. The distributions of transition points and Stokes lines in this case are more complicated than the previous case. There are two limiting cases, b2 >> 1I and b2 << -1, which are similar to the Landau-Zener case. Therefore, the two new formulas for reduced scattering matrix are obtained in these two limiting cases again by simple functions of the two parameters a2 and b2. Especially formula for b2 < -1 is the first one ever obtained. The distributions of transition points and Stokes lines in the region lb2|<1 are very different from and have no correspondence to the former Landau-Zener case. Based on the solvable special differential equation given in chapter 3, an approximate expression for Stokes constant U1 is found with use of a fitting procedure. Again, the a2 - b2 plane is divided into five regions and the best recommended formula for reduced scattering matrix is proposed for each region. Thus, a complete picture of the nonadiabatic tunneling case is attained.