On the solvability in the sense of sequences for some non-Fredholm operators (Q396534)

The authors consider the equation NEWLINE\[NEWLINE-\Delta_xu-\Delta_yu+U(y)u-au=\phi(x,y),\quad x\in\mathbb R^d,\, y\in\mathbb R^3,\, d\in\mathbb N,NEWLINE\]NEWLINE where \(\Delta_x\) and \(\Delta_y\) are the standard Laplace operators acting on the variables \(x\) and \(y\), respectively. They study the corresponding sequence of iterated equations NEWLINE\[NEWLINE-\Delta_xu_n-\Delta_yu_n+U(y)u_n-au_n=\phi_n(x,y),\quad x\in\mathbb R^d,\, y\in\mathbb R^3,\, d\in\mathbb NNEWLINE\]NEWLINE with given orthogonality conditions and show the convergence \(u_n(x,y)\to u(x,y)\) in \(H^2(\mathbb R^{d+3})\) as \(n\to\infty\). The methods of spectral and scattering theory for Schrödinger-type operators are applied.