Abstract
We establish the solvability of certain linear nonhomogeneous equations and demonstrate that under reasonable technical conditions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(L^{2}({\mathbb R}^{d})\) of the solutions. In the first part of the work the equation involves the logarithmic Laplacian. In the second part we generalize the results derived by incorporating a shallow, short-range scalar potential into the problem. The argument relies on the methods of the spectral and scattering theory for the non-Fredholm Schrödinger type operators. As distinct from the preceding articles on the subject, for the operators involved in the equations the essential spectra fill the whole real line.
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The article is dedicated to Professor Roald Sagdeev on the occasion of his 90th birthday.
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Efendiev, M., Vougalter, V. Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian. Monatsh Math 202, 751–771 (2023). https://doi.org/10.1007/s00605-023-01834-1
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DOI: https://doi.org/10.1007/s00605-023-01834-1