Associated LamÉ Equation, Periodic Potentials and sl(2,R)

الأصل · EN

We propose a new approach based on the algebraization of the Associated Lamé equation -ψ''(x) + [m(m+1)k²sn²x + ℓ(ℓ+1)k²(cn²x/dn²x)]ψ(x) = Eψ(x) within sl(2,R) to derive the corresponding periodic potentials. The band edge eigenfunctions and energy spectra are explicitly obtained for integers m,ℓ. We also obtain the explicit expressions of the solutions for half-integer m and integer or half-integer ℓ.