We investigate solvability of certain linear nonhomogeneous elliptic equations and establish that under reasonable technical conditions the convergence in $L^2(R^d)$ of their right sides yields the existence and the convergence in $H^2(R^d)$ of the solutions. The problems involve the sums of second order differential operators without Fredholm property and we apply the methods of spectral and scattering theory for Schroedinger type operators.