Simple high-accuracy method for solving bound-state equations with the Cornell potential in momentum space
Alfred Stadler, Elmar P. Biernat, Vasco Valverde
备注：
14 pages; affiliations updated; 2 new paragraphs and the new Table VII added in Sec. IV; data availability Ref. [40] added; version published in Physical Review D
The well-known Cornell quark-antiquark potential in momentum space contains singularities both in its one-gluon-exchange (OGE) and linear confining parts, which prevents a direct use of the convenient Nyström method to solve the corresponding bound-state integral equation for the meson masses. While it has been known for a long time how the Coulomb-type singularity in the OGE potential can be treated with a subtraction technique, only very complicated methods have been developed to deal with the stronger singularity in the linear potential. In this work, we present a simple subtraction method to remove this singularity from the kernel, such that the Nyström method becomes applicable. Derivatives of the wave function, that appear as a result of the subtraction, are represented by means of interpolating functions, for which we found Lagrange polynomials to be very efficient. Test calculations show excellent agreement with exactly known energy eigenvalues. By increasing the number of integration points and the order of the Lagrange interpolation polynomials, extremely high accuracy can be achieved. This method can also be extended to relativistic Bethe-Salpeter type equations with singular kernels.
On technically solving an effective QCD-Hamiltonian
就其本质而言，场论哈密顿量是在动量表象中导出的。求解相应的积分-微分方程比求解位形空间中较简单的微分方程（“薛定谔方程”）更困难。对于后者，过去已经开发了许多不同且非常有效的方法。但是，而不是傅立叶变换到配置空间-这并不总是容易的-方程在这里直接在动量空间中求解，通过使用高斯求积。特别注意位形空间中的势是线性的，并且相应的动量空间核具有几乎难以处理的1/（\vec k - \vec k '）^4 -奇异性的情况。它的正规化需要一定的技术努力，引入适当的反条款。该方法是数值可靠和快速，比其他方法在文献中。它应该是有用的，也适用于其他方法，包括唯象薛定谔型方程。
