Let L be a non-negative self-adjoint operator acting on L2(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≥2. Let HL1(X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group eitL associated with L such that‖(I+L)−n/2eitLf‖L1(X)+‖(I+L)−n/2eitLf‖HL1(X)≤C(1+|t|)n/2‖f‖HL1(X),t∈R for some constant C=C(n,m)>0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □b on polynomial model domains treated by Nagel–Stein [41], where e−t□b satisfies only the second order Davies-Gaffney estimates.Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint Lp-Sobolev bound for eitL:‖(I+L)−seitLf‖Lp(X)≤C(1+|t|)s‖f‖Lp(X),t∈R,s≥n|12−1p| for every 1<p<∞, which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space Rn.