Perturbations of periodic Sturm–Liouville operators

We study perturbations of the self-adjoint periodic Sturm–Liouville operatorA0=1r0(−ddxp0ddx+q0) and conclude under L1-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.