In this paper, we initiate the study of isogeometric analysis (IGA) of a quantum three-body problem that has been well-known to be difficult to solve. In the IGA setting, we represent the wavefunctions by linear combinations of B-spline basis functions and solve the problem as a matrix eigenvalue problem. The eigenvalue gives the eigenstate energy while the eigenvector gives the coefficients of the B-splines that lead to the eigenstate. The major difficulty of isogeometric or other finite-element-method-based analyses lies in the lack of boundary conditions and a large number of degrees of freedom for accuracy. For a typical many-body problem with attractive interaction, there are bound and scattering states where bound states have negative eigenvalues. We focus on bound states and start with the analysis for a two-body problem. We demonstrate through various numerical experiments that IGA provides a promising technique to solve the three-body problems.
翻译:在本文中,我们开始对已知难以解决的三体量子问题进行等离子分析(IGA) 。 在 IGA 设置中, 我们通过 B- spline 基函数的线性组合来代表波函数, 并将问题作为 egenvaly 的矩阵问题来解决 。 eigenvalu 给出了脑能量, 而 egenvector 给出了导致脑质的B- spline 系数 。 等分数或其他有限元素- 方法分析的主要困难在于缺乏边界条件, 以及高度的准确性自由度。 对于典型的多体性问题, 具有吸引力的互动性, 存在约束状态具有负性等值的交界和分散状态。 我们关注约束状态, 并开始分析二体问题。 我们通过各种数字实验证明 IGA 提供了解决三体问题的有希望的技术 。