一维$δ^{(n)}$函数势的相对论和非相对论薛定谔方程的截断正则化通解及普适性的命运
The General Solution for the Relativistic and Nonrelativistic Schrödinger Equation for the $δ^{(n)}$-Function Potential in 1-dimension Using Cutoff Regularization, and the Fate of Universality
摘要：
本文发展了一种用截断正则化方法求解一维$δ$函数势任意导数的薛定谔方程的一般方法。这项工作处理相对论和非相对论的情况下。在导数$n$是偶数的情况与$n$是奇数的情况之间进行了处理上的区别。导出了每种情况下的一般间隙方程。本文以$δ^{(2)}$函数势为例进行了讨论。结果表明，相对论情形下的$δ^{(2)}$ -函数系统的行为与$δ$ -函数和$δ'$ -函数势完全相同，这意味着它也具有与量子场论相同的特征，如在无质量极限下是渐近自由的，它经历了维数变换，并且具有红外共形不动点.因此，接触相互作用的普遍性的证据已经进一步扩展到包括$δ^{(2)}$功能潜力。
英文摘要：
A general method has been developed to solve the Schrödinger equation for an arbitrary derivative of the $δ$-function potential in 1-d using cutoff regularization. The work treats both the relativistic and nonrelativistic cases. A distinction in the treatment has been made between the case when the derivative $n$ is an even number from the one when $n$ is an odd number. A general gap equations for each case has been derived. The case of $δ^{(2)}$-function potential has been used as an example. The results from the relativistic case show that the $δ^{(2)}$-function system behaves exactly like the $δ$-function and the $δ'$-function potentials, which means it also shares the same features with quantum field theories, like being asymptotically free, in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point. As a result the evidence of universality of contact interactions has been extended further to include the $δ^{(2)}$-function potential.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
