超临界耦合平方反比奇异势散射的J矩阵方法I。理论
J-matrix method of scattering for inverse-square singular potential with supercritical coupling I. Theory
摘要：
散射的J矩阵方法是为了处理规则短程势而发展起来的，在原子、核和分子物理中有着广泛的应用。它的准确性，稳定性和收敛性能与其他成功的散射方法相比毫不逊色。它是一种代数方法，建立在利用满足三项递归关系的正交多项式和三对角矩阵的操作上。最近，我们将该方法扩展到1/r^2奇异短程势的处理，但仅限于1/r^2奇异性的耦合参数强度大于-1/8的亚临界耦合区域。在这项工作中，我们将我们的研究扩展到包括耦合参数强度小于-1/8的超临界耦合。然而，为了实现这一点，我们必须将该方法的公式扩展到满足五项递归关系的对象和五对角矩阵。值得注意的是，我们可以发展的理论没有正则化或自伴扩张，这通常需要在治疗这种高度奇异的潜力。尽管如此，我们不得不付出代价，将方法的公式扩展到更大的代表性中，并应对比通常更慢的收敛。在后续的研究中，我们打算应用该方法来获得散射信息的现实潜力模型。
英文摘要：
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.
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