  In the present paper we consider Schr\"odinger equations with variable
coefficients and potentials, where the principal part is a long-range
perturbation of the flat Laplacian and potentials have at most linear growth at
spatial infinity. We then prove local-in-time Strichartz estimates, outside a
large compact set centered at origin, expect for the endpoint. Moreover we also
prove global-in-space Strichartz estimates under the non-trapping condition on
the Hamilton flow generated by the kinetic energy.
