The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation [Formula: see text] 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 nontrivial potentials allowed being the Coulomb [Formula: see text] potential and the harmonic oscillator [Formula: see text] 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.

中文翻译：

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对偶势与 sl(2) 代数谱问题的交集

某些哈密顿量之间的关系，称为对偶哈密顿量或伙伴哈密顿量，在变换 [公式：见正文] 下，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) 代数的生成元来表达这种哈密顿量来进一步检验这一点，这提供了另一种解决谱问题的方法。似乎这样做极大地限制了允许势的集合，唯一允许的非平凡势是库仑 [公式：见文本] 势和谐振子 [公式：见文本] 势，对于这两者， sl(2)表达式是已知的。看来，通过利用伙伴势变换和量子力学李代数构造的形式，