Solvability in the Sense of Sequences for Some Non-Fredholm Operators in Higher Dimensions

Abstract

We study solvability of certain linear nonhomogeneous elliptic problems and establish that under reasonable technical assumptions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(H^{1}({\mathbb R}^{d})\) of the solutions. The equations involve the square roots of the sums of second order non-Fredholm differential operators and we rely on the methods of the spectral and scattering theory for Schrodinger type operators similarly to our earlier work [26].