Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

Mizutani, Haruya

doi:10.2140/apde.2013.6.1857

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Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

Haruya Mizutani

Vol. 6 (2013), No. 8, 1857–1898

DOI: 10.2140/apde.2013.6.1857

Abstract

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.

Keywords

Schrödinger equation, Strichartz estimates, asymptotically flat metric, unbounded potential, unbounded electromagnetic potentials

Mathematical Subject Classification 2010

Primary: 35B45, 35Q41

Secondary: 35S30, 81Q20

Milestones

Received: 25 February 2012

Revised: 24 September 2012

Accepted: 19 January 2013

Published: 20 April 2014

Authors

Haruya Mizutani
Department of Mathematics, Graduate School of Science Osaka University Toyonaka 560-0043 Japan	Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan

Vol. 6, No. 8, 2013