Singular inverse-square potential: Renormalization and self-adjoint extensions for medium to weak coupling
Singular inverse-square potential: Renormalization and self-adjoint extensions for medium to weak coupling
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DOI：
10.1103/physreva.89.022113
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发表时间：
2014-02
影响因子：
2.9
通讯作者：
Djamil Bouaziz;M. Bawin
中科院分区：
文献类型：
--
作者：
Djamil Bouaziz;M. Bawin
我们研究了质量为 $m$ 的粒子在中弱耦合区域的反平方势 $\alpha/r^{2}$ 场中的径向 Schr\"{o}dinger 方程，即 $-1/4\leq2m\alpha/\hbar^{2}\leq3/4$。通过使用 Beane 的重正化方法 \textit{et} \textit{al.,}具有两个正则化势，一个球形方井和一个球形 $\delta$ 壳，我们说明重正化过程与正则化反项的选择无关。我们证明，在上述耦合常数$\alpha$的范围内，至多存在一个束缚态，与自伴法完全一致 扩展。我们明确表明，这种束缚态是由于重正化方案中存在有吸引力的方阱和δ函数反项所致。我们对 $2m\alpha/\hbar^{2}=-1/4$ 的结果与文献中的一些结果相矛盾。
We study the radial Schr\"{o}dinger equation for a particle of mass $m$ in the field of the inverse-square potential $\alpha/r^{2}$ in the medium-weak-coupling region, i.e., with $-1/4\leq2m\alpha/\hbar^{2}\leq3/4$. By using the renormalization method of Beane \textit{et} \textit{al.,}with two regularization potentials, a spherical square well and a spherical $\delta$ shell, we illustrate that the procedure of renormalization is independent of the choice of the regularization counterterm. We show that, in the aforementioned range of the coupling constant $\alpha$, there exists at most one bound state, in complete agreement with the method of self-adjoint extensions. We explicitly show that this bound state is due to the attractive square-well and delta-function counterterms present in the renormalization scheme. Our result for $2m\alpha/\hbar^{2}=-1/4$ is in contradiction with some results in the literature.