对偶势与SL的相交（2）代数谱问题
The Intersection between Dual Potential and SL(2) Algebraic Spectral Problems
摘要：
在$x{\rightarrow}\bar{x}^{\barα}$变换下，某些哈密顿量之间的关系，称为对偶或伙伴哈密顿量，长期以来一直被用作简化量子力学中谱问题的方法。本文试图通过用SL（2）代数的生成元来表示这种哈密顿算子，进一步研究这一问题，这提供了另一种解决谱问题的方法。这样做似乎极大地限制了允许的势的集合，唯一允许的非平凡势是库仑$\frac{1}{r}$势和谐振子$r^2$势，这两种势的SL（2）表达式都是已知的。它还表明，通过利用伙伴势变换和量子力学的李代数构造的形式主义，有可能从其伙伴哈密顿量的准可解性构造部分哈密顿量的谱。
英文摘要：
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.
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