Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials

Haruya Mizutani

doi:10.3934/cpaa.2014.13.2177

Article Contents

2014, Volume 13, Issue 6: 2177-2210. Doi: 10.3934/cpaa.2014.13.2177

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Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials

Haruya Mizutani^1,

1.
Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588

Received: January 31, 2013

Revised: December 31, 2013

Published: October 2014

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Abstract

In this paper we prove local-in-time Strichartz estimates with loss of derivatives for Schrödinger equations with variable coefficients and potentials, under the conditions that the geodesic flow is nontrapping and potentials grow polynomially at infinity. This is a generalization to the case with variable coefficients and improvement of the result by Yajima-Zhang [40]. The proof is based on microlocal techniques including the semiclassical parametrix for a time scale depending on a spatial localization and the Littlewood-Paley type decomposition with respect to both of space and frequency.

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Mathematics Subject Classification: Primary 35Q41; Secondary 35B45.

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