  We prove a sharp H\"ormander multiplier theorem for Schr\"odinger operators
$H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition
on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for
$H$, which is a weaker condition than that for nonnegative operators via the
heat kernel approach. Our approach is elaborated in one dimension with
potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we
assume that $\int (1+|x|) |V(x)|dx$ is finite and $H$ has no resonance at zero.
In the resonance case we assume $\int (1+|x|^2) |V(x)| dx$ is finite.
