Perturbation and spectral theory for singular indefinite Sturm-Liouville operators. (English) Zbl 07901384
Summary: We study singular Sturm-Liouville operators of the form
\[
\frac{1}{r_j} \left(-\frac{\operatorname{d}}{\operatorname{d} x} p_j \frac{\operatorname{d}}{\operatorname{d} x}+q_j \right), \quad j=0,1,
\]
in \(L^2 ((a, b); r_j)\) with endpoints \(a\) and \(b\) in the limit point case, where, in contrast to the usual assumptions, the weight functions \(r_j\) have different signs near \(a\) 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 \(L^1\) perturbations of periodic operators.
MSC:
| 34B24 | Sturm-Liouville theory |
| 34L05 | General spectral theory of ordinary differential operators |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
Keywords:
indefinite Sturm-Liouville operators; perturbations; relative oscillation; essential spectrum; discrete spectrum; periodic coefficientsReferences:
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