SOLVABILITY IN THE SENSE OF SEQUENCES FOR SOME NON-FREDHOLM OPERATORS WITH THE LOGARITHMIC LAPLACIAN

: We establish the solvability of certain linear nonhomogeneous equations and demonstrate that under reasonable technical conditions the convergence in L 2 ( R d ) of their right sides implies the existence and the convergence in L 2 ( 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 Schr¨odinger 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.