  We obtain the band edge eigenfunctions and the eigenvalues of solvable
periodic potentials using the quantum Hamilton - Jacobi formalism. The
potentials studied here are the Lam{\'e} and the associated Lam{\'e} which
belong to the class of elliptic potentials. The formalism requires an
assumption about the singularity structure of the quantum momentum function
$p$, which satisfies the Riccati type quantum Hamilton - Jacobi equation, $
p^{2} -i \hbar \frac{d}{dx}p = 2m(E- V(x))$ in the complex $x$ plane. Essential
use is made of suitable conformal transformations, which leads to the
eigenvalues and the eigenfunctions corresponding to the band edges in a simple
and straightforward manner. Our study reveals interesting features about the
singularity structure of $p$, responsible in yielding the band edge
eigenfunctions and eigenvalues.
