In this note, we study the potential algebra for several models arising out of quantum mechanics with generalized uncertainty principle. We first show that the eigenvalue equation corresponding to the momentum-space Hamiltonian \[H=-(1+\beta p^{ 2})\frac{ d}{ dp}(1+\beta p^{ 2})\frac{ d}{ dp}+g(g-1)\beta^{ 2}p^{ 2}-g\beta,\] which is associated with some one-dimensional models with minimal length uncertainty, can be solved by the unitary representations of the Lie algebra $\mathfrak{ su}(2)$ if $g\in\{ \tfrac{ 1}{ 2},1,\tfrac{ 3}{ 2},2,\cdots\}$. We then apply this result to spectral problems for the non-relativistic harmonic oscillator as well as the relativistic Dirac oscillator in the presence of a minimal length and show that these problems can be solved solely in terms of $\mathfrak{ su}(2)$.
Categories: quant-ph hep-th math-ph math.MP nlin.SI