库仑势的一维散射问题、自伴扩张、重整化和点$δ$相互作用
One-dimensional scattering problem, self-adjoint extensions, renormalizations and point $δ$-interactions for Coulomb potential
摘要：
本文讨论了初始势为$α|x|^{-1}$的一维单中心散射问题，并将其归结为寻找允许的自伴扩张。利用守恒定律作为奇点的必要条件，并考虑到基本解的解析结构，我们可以得到波函数（即边界条件）、散射系数和对势的奇异修正以及相应的束缚态谱的精确表达式。结果表明，在任意的自伴延拓中，对势的点$δ$形修正都是存在的，而且这种修正的形式与量子电动力学中的重整化项的形式相对应.因此，所提出的方法显示了明确的边界条件，散射系数和$δ$形添加到潜在的连接。作为一个整体，该方法展示了在分析适当的Hamilton算子的自伴扩展时出现的机会。由于它涉及到重整化理论，该方法可以看作是Bogoliubov，Parasiuk和Hepp重整化方法的推广。
英文摘要：
In the paper the one-dimensional one-center scattering problem with the initial potential $α|x|^{-1}$ on the whole axis is treated and reduced to the search for allowable self-adjoint extensions. Using the laws of conservation as necessary conditions in the singular point alongside with account of the analytical structure of fundamental solutions, it allows us to receive exact expressions for the wave functions (i.e. for the boundary conditions), scattering coefficients and the singular corrections to the potential, as well as the corresponding bound state spectrum. It turns out that the point $δ$-shaped correction to the potential should be present without fail at any choice of the allowable self-adjoint extension, moreover a form of these corrections corresponds to the form of renormalization terms obtained in quantum electrodynamics. Thus, the proposed method shows the unequivocal connection among the boundary conditions, scattering coefficients and $δ$-shaped additions to the potential. Taken as a whole, the method demonstrates the opportunities which arise at the analysis of the self-adjoint extensions of the appropriate Hamilton operator. And as it concerns the renormalization theory, the method can be treated as a generalization of the Bogoliubov, Parasiuk and Hepp method of renormalizations.
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