Let \((-\Delta )^\frac{\alpha }{2}+|x|^a\) be a generalized Schrödinger operator on the Euclidean space \(\mathbb {R}^n\) with \(\alpha \in (0,n)\) and \(a\in (-\infty ,+\infty )\) . In this article, we show that \(|x|^a\) is infinitesimally relatively bounded with respect to \((-\Delta )^\frac{\alpha }{2}\) in \(L^p(\mathbb {R}^n)\) with \(p\in (1,\infty )\) if and only if \(a\in (-\min \{n/p,\alpha \},0]\) . A key novelty here is the reduction of this question to verifying a specific Carleson condition through the application of sparse domination technique. Moreover, similar results are obtained for Schrödinger operator \(-\Delta +\mathop \mathrm {\,dist\,}(x,\partial \Omega )^a\) on a bounded domain \(\Omega \) .
