Separation for Schr\"odinger operators. A counterexample to Simon's conjecture

Abstract

We construct, for every $d\ge 5$, a nonnegative radial potential $V\in L^\infty_{\mathrm{loc}}(\mathbb{R}^d\setminus\{0\})\cap L^1(\mathbb{R}^d)$ for which $V(-\Delta\dotplus V)^{-1}$ is not bounded on $L^2(\mathbb{R}^d)$.