Supersymmetry and solvable periodic potentials

We use the formalism of supersymmetric quantum mechanics to enlarge considerably the limited class of analytically solvable one-dimensional periodic potentials. In particular, we derive and discuss the energy-band structure of the Lamé potentials pm sn² (x, m) and associated Lamé potentials pm sn² (x, m) + qm cn²(x, m)/dn² (x, m), both of which involve Jacobi elliptic functions with modulus parameter m. We find several new analytic expressions for band-edge energies and wave functions. The supersymmetric partners of Lamé and associated Lamé potentials constitute even more new solvable potentials with exactly the same energy-band structure.