Perturbations of periodic Sturm-Liouville operators

We study perturbations of self-adjoint periodic Sturm-Liouville operators and conclude under L1-assumptions on the differences of the coeffcients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coeffcients, 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.


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