We propose a new approach based on the algebraization of the Associated Lam\'{ e} equation \[-\psi''(x) + [ m(m+1)k^{ 2}sn^{ 2}x + \ell(\ell+1)k^{ 2}(cn^{ 2}x/dn^{ 2}x)]\psi(x) = E\psi(x)\] within sl(2,R) to derive the corresponding periodic potentials. The band edge eigenfunctions and energy spectra are explicitly obtained for integers m,$\ell$. We also obtain the explicit expressions of the solutions for half-integer m and integer or half-integer $\ell$.
Categories: math-ph hep-th math.MP quant-ph