  Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$
is a space of homogeneous type with a dimension $n$. Suppose that the heat
kernel of $L$ satisfies the Davies-Gaffney estimates of order $m\geq 2$. Let
$H^1_L(X)$ be the Hardy space associated with $L.$ In this paper we show sharp
endpoint estimate for the Schr\"odinger group $e^{itL}$ associated with $L$
such that
  \begin{eqnarray*}
  \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ L^1(X)} + \left\|
(I+L)^{-{n/2}}e^{itL} f\right\|_{ H^1_L(X)}
  \leq C(1+|t|)^{n/2}\|f\|_{H^1_L(X)}, \ \ \ t\in{\mathbb R} \end{eqnarray*}
for some constant $C=C(n, m)>0$ independent of $t$. By a duality and
interpolation argument, it gives a new proof of a recent result of \cite{CDLY}
for { sharp} endpoint $L^p$-Sobolev bound for $e^{itL}$: $$
  \left\| (I+L)^{-s }e^{itL} f\right\|_{ L^p(X)} \leq C (1+|t|)^{s} \|f\|_{
L^p(X)}, \ \ \ t\in{\mathbb R}, \ \
  \ s\geq n\big|{1\over 2}-{1\over p}\big| $$ for every $1<p<\infty$ when the
heat kernel of $L$ satisfies a Gaussian upper bound, which extends the
classical results due to Miyachi ) for the Laplacian on the Euclidean space
${\mathbb R}^n$.
