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Solvability in the sense of sequences for some logarithmic Schrödinger operators in higher dimensions

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Abstract

We study the solvability of certain linear nonhomogeneous equations containing the logarithm of the sum of the two Schrödinger operators in higher dimensions and demonstrate that under the reasonable technical assumptions the convergence in \(L^{2}({{\mathbb {R}}}^{d})\) of the right sides yields the existence and the convergence in \(L^{2}({{\mathbb {R}}}^{d})\) of the solutions. The equations involve the operators without the Fredholm property and we use the methods of the spectral and scattering theory for the Schrödinger type operators to generalize the results of our preceding work Efendiev and Vougalter(Monatsh. Math., 2023). As distinct from the many previous articles on the subject, for the operators contained in our equations the essential spectra fill the whole real line.

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Acknowledgements

The authors express their gratitude to the anonymous referee for the useful remarks.

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Authors and Affiliations

  1. Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstrasse 1, 85764, Neuherberg, Germany

    Messoud Efendiev

  2. Department of Mathematics, Marmara University, Istanbul, Turkey

    Messoud Efendiev

  3. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Vitali Vougalter

Authors
  1. Messoud Efendiev
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  2. Vitali Vougalter
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The corresponding author has read the Springer journal policies on author responsibilities and submits this manuscript in accordance with those policies. ME and VV wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Vitali Vougalter.

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Efendiev, M., Vougalter, V. Solvability in the sense of sequences for some logarithmic Schrödinger operators in higher dimensions. J. Pseudo-Differ. Oper. Appl. 14, 32 (2023). https://doi.org/10.1007/s11868-023-00527-5

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  • DOI: https://doi.org/10.1007/s11868-023-00527-5

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