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Perturbation and spectral theory for singular indefinite Sturm-Liouville operators

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

We study singular Sturm-Liouville operators of the form 1/r_j (- d/ dx p_j d/ dx +q_j),   j=0,1, in L²((a,b);r_j), where, in contrast to the usual assumptions, the weight functions r_j have different signs near the singular endpoints a and b. In this situation the associated maximal operators become self-adjoint with respect to indefinite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights r_j are allowed. Special attention is paid to Kneser type results in the indefinite setting and to L¹ perturbations of periodic operators.

Keywords: indefinite Sturm-Liouville operators, perturbations, relative oscillation, essential spectrum, discrete spectrum, periodic coefficients

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