具有衰减势的三体薛定谔方程
On the three-body Schrödinger equation with decaying potentials
摘要：
证明了在平方可积函数空间中的三体Schr dinger算子是生成指数酉群的算子的某种推广，该酉群包含一个长度为$κ+1$，$κ=0,1,...$的幂零李代数。结果证明了具有衰减势的三体薛定谔方程的解存在于换位子代数中。对于库仑三体系统，我们的任务是求解--在这些子代数中--三维径向薛定谔方程，其反幂势形式为$r^{-κ-1}$。作为对库仑体系的应用，给出了某些下界态的解析解。在与三单位电荷系统有关的条件下，用$κ=0$得到的解，在阈值处被减少到众所周知的束缚态的本征值。
英文摘要：
The three-body Schrödinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length $κ+1$, $κ=0,1,...$ As a result, the solutions to the three-body Schrödinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve - in these subalgebras - the radial Schrödinger equation in three dimensions with the inverse power potential of the form $r^{-κ-1}$. As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with $κ=0$, are reduced to the well-known eigenvalues of bound states at threshold.
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