  The relation between certain Hamiltonians, known as dual, or partner
Hamiltonians, under the transformation $x{\rightarrow}\bar{x}^{\bar{\alpha}}$
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.
