论文标题
J-Matrix的散射方法，用于与超临界耦合I的逆向奇异电位。理论
J-matrix method of scattering for inverse-square singular potential with supercritical coupling I. Theory
论文作者
论文摘要
开发了J-Matrix的J-Matrix方法，以处理适用于原子，核和分子物理学的常规短距离电位。它的准确性，稳定性和收敛性与其他成功的散射方法相比。它是一种代数方法，它建立在满足三项递归关系和操纵三角形矩阵的正交多项式的利用之上。最近，我们将方法扩展到1/r^2奇异的短距离电位的处理，但将自己局限于亚临界耦合方案，其中1/r^2奇异性的耦合参数强度大于-1/8。在这项工作中，我们将研究扩展到包括耦合参数强度小于-1/8的超临界耦合。但是，为此，我们必须将方法的表述扩展到满足五项递归关系和矩阵的对象。值得注意的是，我们可以在没有正则化或自我拥护者扩展的情况下发展理论，这通常是在处理如此高度奇异的电位时所需的。尽管如此，我们必须通过将方法的表述扩展到更大的表示形式，并应对比通常的收敛速度较慢来付出代价。在一项后续研究中，我们打算采用该方法来获取现实潜在模型的散射信息。
The J-matrix method of scattering was developed to handle regular short-range potentials with applications in atomic, nuclear and molecular physics. Its accuracy, stability, and convergence properties compare favorably with other successful scattering methods. It is an algebraic method, which is built on the utilization of orthogonal polynomials that satisfy three-term recursion relations and on the manipulation of tridiagonal matrices. Recently, we extended the method to the treatment of 1/r^2 singular short-range potentials but confined ourselves to the sub-critical coupling regime where the coupling parameter strength of the 1/r^2 singularity is greater than -1/8. In this work, we expand our study to include the supercritical coupling in which the coupling parameter strength is less than -1/8. However, to accomplish that we had to extend the formulation of the method to objects that satisfy five-term recursion relations and matrices that are penta-diagonal. It is remarkable that we could develop the theory without regularization or self-adjoint extension, which are normally needed in the treatment of such highly singular potentials. Nonetheless, we had to pay the price by extending the formulation of the method into this larger representation and by coping with slower than usual convergence. In a follow-up study, we intend to apply the method to obtain scattering information for realistic potential models.
