Berggren基下长程库仑势的数值处理
Numerical treatment of long-range Coulomb potential with Berggren bases
摘要：
包含长程库仑势的薛定谔方程采用Fredholm方程的形式，当由承载连续状态的基表示时，其核在其对角线上是奇异的，例如在傅里叶-贝塞尔变换中。已经设计了几种方法来解决这个困难，从简单地删除的库仑势的无限范围与屏蔽或切割功能，使用离散计划，利用库仑核奇异性的可积性。然而，他们从来没有被测试的背景下，Berggren基地，允许多体核波函数被扩大，与晕或共振性质的壳模型框架内。因此，本文的目的是测试不同的离散方案的库仑势核的框架内复杂的能量核物理。为此，与A ~ 20区域中产生的sd壳层相关的质子态的Berggren基展开将受到影响，通常是共振的。除了标准的框架，涉及到一个削减功能或分析积分的奇异性，一个新的方法，它取代了有限的非对角项的对角奇异性。它将被证明，这种方法在精度上优于前两种技术。
英文摘要：
The Schrodinger equation incorporating the long-range Coulomb potential takes the form of a Fredholm equation whose kernel is singular on its diagonal when represented by a basis bearing a continuum of states, such as in a Fourier-Bessel transform. Several methods have been devised to tackle this difficulty, from simply removing the infinite-range of the Coulomb potential with a screening or cut function to using discretizing schemes which take advantage of the integrable character of Coulomb kernel singularities. However, they have never been tested in the context of Berggren bases, which allow many-body nuclear wave functions to be expanded, with halo or resonant properties within a shell model framework. It is thus the object of this paper to test different discretization schemes of the Coulomb potential kernel in the framework of complex-energy nuclear physics. For that, the Berggren basis expansion of proton states pertaining to the sd-shell arising in the A ~ 20 region, being typically resonant, will be effected. Apart from standard frameworks involving a cut function or analytical integration of singularities, a new method will be presented, which replaces diagonal singularities by finite off-diagonal terms. It will be shown that this methodology surpasses in precision the two former techniques.
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