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Bounds on the non-real spectrum of differential operators with indefinite weights

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Abstract

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and \(\infty \) are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.

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Notes

  1. Definition 2.1 is slightly more general than the definition used in [10, Definition 3.1]. Contrary to the definition in [10], a spectral projector for an operator non-negative over \(\overline{\mathbb{C }}{\setminus } K\) in the sense of Definition 2.1 corresponding to the set \(\overline{\mathbb{C }}{\setminus } K\) does, in general, not exist.

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Acknowledgments

The authors wish to thank the anonymous referee for helpful comments and remarks that led to an improvement of the manuscript. The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 , Graz, Austria

    Jussi Behrndt

  2. Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17. Juni 136, 10623 , Berlin, Germany

    Friedrich Philipp

  3. Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 , Ilmenau, Germany

    Carsten Trunk

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Correspondence to Jussi Behrndt.

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Behrndt, J., Philipp, F. & Trunk, C. Bounds on the non-real spectrum of differential operators with indefinite weights. Math. Ann. 357, 185–213 (2013). https://doi.org/10.1007/s00208-013-0904-7

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