A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$

We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $\mbox{GL}_2({\mathbb C}[h])$ and rely on Cohn's standard form.