In the present paper we consider Schrödinger 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, without loss of derivatives. Moreover we also prove global-in-space Strichartz estimates under the non-trapping condition on the Hamilton flow generated by the kinetic energy.
Haruya MIZUTANI. "Strichartz estimates for Schrödinger equations with variable coefficients and potentials at most linear at spatial infinity." J. Math. Soc. Japan 65 (3) 687 - 721, July, 2013. https://doi.org/10.2969/jmsj/06530687