In this expository paper, we describe a sequence of earlier papers presenting applications of a general theorem regarding pointwise estimates for kernels of Neumann series operators \(\sum _{j=0}^{\infty } T^j\) . Here T is an integral operator with a quasi-metric kernel on a measure space \((\Omega , \omega )\) , with \(\Vert T \Vert _{L^2(\omega ) \rightarrow L^2 (\omega )} <1\) . Applications are made to the study of non-negative solutions u to the time-independent Schrödinger equation \(- \triangle u = qu\) on a domain \(\Omega \subseteq \mathbb {R}^n, n \ge 3\) , with \(u = f \) on \(\partial \Omega \) , where \(q \in L^1_{{\textit{loc}}}(\Omega )\) and q and f are non-negative. We obtain a balayage condition on the potential q measuring how rapidly q can blow up at \(\partial \Omega \) and still allow for an almost everywhere finite solution. We also derive bilateral estimates for the Green’s function and Poisson kernel for the Schrödinger operator \(-\triangle -q\) in terms of q and the Green’s function and Poisson kernel for the Laplacian. These results are first described for a \(C^2\) domain. They are later extended to analogues involving the Martin kernel and harmonic measure on a uniform domain.
