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

Abstract

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

8 pages, no figure, tex file(version 2.09)

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