Klein佯谬与半经典Dirac方程的散射理论
Klein paradox and Scattering theory for the semi-classical Dirac equation
摘要：
我们研究了半经典Dirac算子在$\R$上的Klein佯谬，其中势函数在无穷远处具有常数极限，但不一定相同。利用复WKB方法，建立了以入射和出射平面波解表示的与时间无关的散射理论。相应的散射矩阵是酉矩阵。我们得到了关于半经典参数$h$的散射矩阵在Klein佯谬、全透射和全反射情况下的渐近展开式。最后，我们在零质量的情况下处理散射问题。
英文摘要：
We study the Klein paradox for the semi-classical Dirac operator on $\R$ with potentials having constant limits, not necessarily the same at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter $h$, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.
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