论文标题
在球体上的开普勒 - 库仑电势的准精细解决的扩展
Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere
论文作者
论文摘要
我们考虑在$ d $维球体上的开普勒 - 库仑电位的扩展家族，并在变形的超对称框架中进行分析，其中众所周知，起始电位显示出变形的形状不变性属性。我们表明，大家庭的成员也被赋予了这样的属性，只要某些约束条件就可以满足潜在参数，换句话说，它们是有条件变形的形状不变的。由于在构建合作伙伴潜在层次结构的第二步中，约束条件发生了变化，因此我们在两组之间施加了兼容条件，以构建具有已知地面和首先兴奋的状态的准上可解决的潜力。为家族的前三名成员获得了一些明确的结果。然后，我们使用生成函数方法，其中前两个超势，前两个伙伴电位和起始电位的前两个特征态是由某些生成函数$ w _+（r）$ [及其随附的函数$ w _-（r）$构建的。从前三个家庭成员的后者获得的结果中，我们提出了一些$ W _ {\ pm}（r）$对$ m $ th家庭成员有效的公式，具体取决于$ m+1 $ $ a_0 $，$ a_1 $，$ a_1 $，\ ldots，$ a_m $ $ a_m $。这样的常数满足$ M+1 $线性方程的系统。解决后者使我们能够将结果扩展到第七个家庭成员，然后提出一个猜想，从而从问题的参数方面给出了$ a_i $常数的一般结构。
We consider a family of extensions of the Kepler-Coulomb potential on a $d$-dimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property. We show that the members of the extended family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build quasi-exactly solvable potentials with known ground and first-excited states. Some explicit results are obtained for the first three members of the family. We then use a generating function method, wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function $W_+(r)$ [and its accompanying function $W_-(r)$]. From the results obtained for the latter for the first three family members, we propose some formulas for $W_{\pm}(r)$ valid for the $m$th family member, depending on $m+1$ constants $a_0$, $a_1$, \ldots, $a_m$. Such constants satisfy a system of $m+1$ linear equations. Solving the latter allows us to extend the results up to the seventh family member and then to formulate a conjecture giving the general structure of the $a_i$ constants in terms of the parameters of the problem.
