库仑三体问题中的超球库仑椭球表象
Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem
摘要：
利用可分库仑两中心问题$ϕ_j(ξ,η)=X_j(ξ)Y_j(η)$的已知解，得到了库仑三体波函数的新表示，其中$X_j(ξ)$和$Y_j(η)$是库仑球函数.它的显著特点是与三粒子破碎下散射问题的边界条件相协调。也就是说，任何开放通道中散射粒子的波函数是与该通道对应的单个项的渐近性，该项是所建议的展开项。这一效果是由于建立了三体系统三个内坐标与$ϕ_j(ξ,η)$参数之间的新关系。它确保了$ϕ_j(ξ,η)$在恒定超半径的球体上的正交性，$ρ=const$，代替了传统Born-Oppenheimer方法中出现的表面$R=|\bi{x}_2-\bi{x}_1|=const$。自变量$ξ$和$η$是在该球面上的正交坐标，其中三个极点在聚结点中。用赤平投影将它们与平面上的椭圆坐标联系起来。对于总角动量$J\ge 0$，$ϕ_j$和Wigner $D$函数的乘积形成了在五维超球面上的超球面库仑椭球（HSCS）基，$ρ$是一个参数。给出了三体波函数HSCS分解系数径向函数$f^J_i(ρ)$的微分方程组和边界条件。
英文摘要：
The new representation of the Coulomb three-body wave function via the well-known solutions of the separable Coulomb two-centre problem $ϕ_j(ξ,η)=X_j(ξ)Y_j(η)$ is obtained, where $X_j(ξ)$ and $Y_j(η)$ are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the boundary conditions of the scattering problem below the three-particle breakup. That is, the wave function of the scattering particles in any open channel is the asymptotics of the single, corresponding to that channel, term of the expansion suggested. The effect is achieved due to the new relation between three internal coordinates of the three-body system and the parameters of $ϕ_j(ξ,η)$. It ensures the orthogonality of $ϕ_j(ξ,η)$ on the sphere of a constant hyperradius, $ρ=const$, in place of the surface $R=|\bi{x}_2-\bi{x}_1|=const$ appearing in the traditional Born-Oppenheimer approach. The independent variables $ξ$ and $η$ are the orthogonal coordinates on that sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of the stereographic projection. For the total angular momentum $J\ge 0$ the products of $ϕ_j$ and the Wigner $D$-functions form the hyperspherical Coulomb spheroidal (HSCS) basis on the five-dimensional hypersphere, $ρ$ being a parameter. The system of the differential equations and the boundary conditions for the radial functions $f^J_i(ρ)$, the coefficients of the HSCS decomposition of the three-body wave function, are presented.
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