奇异势与湮灭
Singular potentials and annihilation
摘要：
本文讨论了奇异吸引势$-α_{s}/r^{s}$、$s\geq 2$的正则化问题，通过对相互作用常数$α_{s}=α_{s}\pm i0$进行无穷小虚加.这样的过程使得散射可观测量的唯一定义成为可能，并且等于原点中粒子的吸收（产生）。结果表明，所建议的正则化方法是相互作用常数为$α_{s}$的排斥奇异势散射振幅的解析延拓。研究了用这种方法正则化的奇异势的近阈值性质。我们得到的散射长度的表达式，它变成复杂的，即使是无穷小的相互作用常数的虚部。本文讨论了正则势近阈态受奇异势微扰的问题，得到了能级位移和宽度的表达式。我们表明，建议的正则化的物理意义是，散射观测值是不敏感的任何细节的奇异位的短程修改，如果存在足够强的非弹性短程相互作用。在这种情况下，散射观测量由正则化势$-(α_{s}\pm i0)/r^{s}$的薛定谔方程的解确定。我们指出，发展的形式主义可以适用于短程湮灭系统的描述，特别是低能核子-反核子散射。
英文摘要：
We discuss the regularization of attractive singular potentials $-α_{s}/r^{s}$, $s\geq 2$ by infinitesimal imaginary addition to interaction constant $α_{s}=α_{s}\pm i0$. Such a procedure enables unique definition of scattering observables and is equal to an absorption (creation) of particles in the origin. It is shown, that suggested regularization is an analytical continuation of the scattering amplitudes of repulsive singular potential in interaction constant $α_{s}$. The nearthreshold properties of regularized in a mentioned way singular potential are examined. We obtain expressions for the scattering lengths, which turn to be complex even for infinitesimal imaginary part of interaction constant. The problem of perturbation of nearthreshold states of regular potential by a singular one is treated, the expressions for level shifts and widths are obtained. We show, that the physical sense of suggested regularization is that the scattering observables are insensitive to any details of the short range modification of singular potential, if there exists sufficiently strong inelastic short range interaction. In this case the scattering observables are determined by solutions of Schrodinger equation with regularized potential $-(α_{s}\pm i0)/r^{s}$. We point out that the developed formalism can be applied for the description of systems with short range annihilation, in particular low energy nucleon-antinucleon scattering.
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