Article
Adv. Math. 422, 109022, 22 pp (2023) [DOI: 10.1016/j.aim.2023.109022]

Perturbations of periodic Sturm-Liouville operators

Jussi Behrndt, Philipp Schmitz, Gerald Teschl, and Carsten Trunk

Abstract
We study perturbations of the self-adjoint periodic Sturm-Liouville operator
A0 = 1/r0 (- d/ dx p0 d/ dx + 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.

MSC2020: Primary 34L05, 81Q10; Secondary 34L40, 47E05
Keywords: Periodic Sturm-Liouville operators, perturbations, essential spectrum, absolutely continuous spectrum, spectral gaps, discrete eigenvalues

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