The aim of this paper is to determine conditions on a potential \(V\) which ensure that an operator \(H:=H_0+V\) acting on \(L^2(\mathbb{R}^N)\) defines a semigroup acting in \(L^p(\mathbb{R}^N)\). Here \(H_0\) is an operator such as \((-\Delta)^m\) which is self-adjoint, uniformly elliptic, of homogeneous order \(2m\), with heat kernel satisfying appropriate bounds. For example, if \(N<2m\) and \(V\in L^1 \), then \(H\) generates a holomorphic semigroup on \(L^1\). On the other hand, if \(N >2m\) and \[ \lim_{\delta\to 0}\sup_{x\in R^N}\int|x-y|^{2m-N}|V(y) |dy=0, \] then \(V\) has zero relative bound in \(L^1\) with respect to \(H_0\). Moreover, the operator \(e^{-Ht}\) is bounded from \(L^p\) to \(L^q\) for all \(1\leq p\leq q\leq \infty\) and all \(t\geq 0\).