Nonlocal supersymmetric deformations of periodic potentials

Irreducible second-order Darboux transformations are applied to the periodic Schrödinger operators. It is shown that for the pairs of factorization energies inside the same forbidden band they can create new nonsingular potentials with periodicity defects and bound states embedded in the spectral gaps. The method is applied to the Lamé and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.