Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials. (English) Zbl 1297.35061

Summary: This paper is concerned with Schrödinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (respectively magnetic) potential can grow subquadratically (respectively sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at the origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. Finally, under the nontrapping condition, we prove Strichartz estimates with an arbitrarily small derivative loss without asymptotic flatness on the coefficients.


35B45 A priori estimates in context of PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35S30 Fourier integral operators applied to PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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