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 differences coefficients that essential spectrum absolutely continuous remain same. If a finite first moment condition holds for coefficients, then at most finitely many eigenvalues appear in spectral gaps. This observation extends seminal result by Rofe-Beketov from 1960s. Finally, imposing second we show band edges are no perturbed operator.