Messoud Efendiev, Vitali Vougalter
Solvability in the sense of sequences for some non-Fredholm operators 
with the logarithmic Laplacian
(194K, pdf)

ABSTRACT.  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 Schrodinger 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.