奇异吸引$1/r^4$势的重整化
Renormalization of the singular attractive $1/r^4$ potential
摘要：
本文研究了质量为$m$的粒子在奇异吸引$g^2/{r^4}$势场中的径向薛定谔方程，特别强调了束缚态问题。利用Beane \textit{et al.}的正则化方法，解析求解了相应的"重整化群流”方程.我们发现与以前的研究一致，它的解决方案具有极限环行为，并有无限多个分支。我们表明，一个连续的选择的解决方案对应于一个给定的固定数量的束缚态和低能量的相移不断变化的能量。我们详细研究了这种正则化方法和传统的方法修改的短程部分的潜力与无限排斥硬核之间的联系。我们表明，这两种方法产生的束缚态的结果接近一致，即使正则化方法的比恩\textit{et al.}不包括明确的问题中的任何新的规模。进一步说明了正则化方法在计算无偶极矩中性极化分子场的电子束缚态中的应用。我们发现的结合能的s-波极化束缚电子在该领域的C $_{60}$分子为17毫电子伏的散射长度对应于一个硬核半径的大小的分子半径（$\sim 3.37$）。这一结果可以进一步与最近的两个参数拟合使用的Lennard-Jones潜在产生的结合能范围从3到25毫电子伏。
英文摘要：
We study the radial Schrödinger equation for a particle of mass $m$ in the field of a singular attractive $g^2/{r^4}$ potential with particular emphasis on the bound states problem. Using the regularization method of Beane \textit{et al.}, we solve analytically the corresponding ``renormalization group flow" equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound states results in close agreement even though the regularization method of Beane \textit{et al.} does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of s-wave polarization bound electrons in the field of C$_{60}$ molecules to be 17 meV for a scattering length corresponding to a hard core radius of the size of the molecule radius ($\sim 3.37$ Å). This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.
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