论文标题
双电势与SL（2）代数光谱问题之间的交点
The Intersection between Dual Potential and SL(2) Algebraic Spectral Problems
论文作者
论文摘要
在转换下，某些被称为双重的汉密尔顿人或伴侣哈密顿人之间的关系$ x {\ rightarrow} \ bar {x}^{\barα} $长期以来一直用作简化量子力学中光谱问题的方法。本文旨在通过SL（2）代数的发电机来表达此类汉密尔顿人，从而进一步研究这一点，该代数提供了解决频谱问题的另一种方法。看来，这样做很大程度上限制了一组允许的电位，唯一的非平凡潜力是库仑$ \ frac {1} {r} $势和谐波振荡器$ r^2 $潜力，对于SL（2）的表达式已经知道。同样，通过利用伙伴的潜在转型和量子力学的谎言代数构建的形式主义，可以通过其伴侣汉密尔顿的准可溶性来构建哈密顿量的一部分。
The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation $x{\rightarrow}\bar{x}^{\barα}$ has long been used as a method of simplifying spectral problems in quantum mechanics. This paper seeks to examine this further by expressing such Hamiltonians in terms of the generators of SL(2) algebra, which provides another method of solving spectral problems. It appears that doing so greatly restricts the set of allowable potentials, with the only non-trivial potentials allowed being the Coulomb $\frac{1}{r}$ potential and the Harmonic Oscillator $r^2$ potential, for both of which the SL(2) expression is already known. It also appears that, by utilizing both the partner potential transformation and the formalism of the Lie-algebraic construction of quantum mechanics, it may be possible to construct part of a Hamiltonian's spectrum from the quasi-solvability of its partner Hamiltonian.
