  Let $L = \Delta + V$ be Schr{\"o}dinger operator with a non-negative
potential $V$ on a complete Riemannian manifold $M$. We prove that the conical
square functional associated with $L$ is bounded on $L^p$ under different
assumptions. This functional is defined by $$ \mathcal{G}_L (f) (x) = \left(
\int_0^\infty \int_{B(x,t^{1/2})} |\nabla e^{-tL} f(y)|^2 + V |e^{-tL} f(y)|^2
\frac{\mathrm{d}t \mathrm{d}y}{Vol(y,t^{1/2})} \right)^{1/2}.$$For $p \in
[2,+\infty)$ we show that it is sufficient to assume that the manifold has the
volume doubling property whereas for $p \in (1,2)$ we need extra assumptions of
$L^p-L^2$ of diagonal estimates for $\{ \sqrt{t} \nabla e^{-tL}, t\geq 0 \}$
and $ \{ \sqrt{t} \sqrt{V} e^{-tL} , t \geq 0\}$.Given a bounded holomorphic
function $F$ on some angular sector, we introduce the generalized conical
vertical square functional$$\mathcal{G}_L^F (f) (x) = \left( \int_0^\infty
\int_{B(x,t^{1/2})} |\nabla F(tL) f(y)|^2 + V |F(tL) f(y)|^2 \frac{\mathrm{d}t
\mathrm{d}y}{Vol(y,t^{1/2})} \right)^{1/2}$$ and prove its boundedness on $L^p$
if $F$ has sufficient decay at zero and infinity. We also consider conical
square functions associated with the Poisson semigroup, lower bounds, and make
a link with the Riesz transform.
