Vol. 6, No. 8, 2013

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Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

Haruya Mizutani

Vol. 6 (2013), No. 8, 1857–1898

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.

Schrödinger equation, Strichartz estimates, asymptotically flat metric, unbounded potential, unbounded electromagnetic potentials
Mathematical Subject Classification 2010
Primary: 35B45, 35Q41
Secondary: 35S30, 81Q20
Received: 25 February 2012
Revised: 24 September 2012
Accepted: 19 January 2013
Published: 20 April 2014
Haruya Mizutani
Department of Mathematics, Graduate School of Science
Osaka University
Toyonaka 560-0043
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502