Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 8 (2013), 1857-1898.

Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

Haruya Mizutani

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 6, Number 8 (2013), 1857-1898.

Dates
Received: 25 February 2012
Revised: 24 September 2012
Accepted: 19 January 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731448

Digital Object Identifier
doi:10.2140/apde.2013.6.1857

Mathematical Reviews number (MathSciNet)
MR3198586

Zentralblatt MATH identifier
1297.35061

Subjects
Primary: 35B45: A priori estimates 35Q41: Time-dependent Schrödinger equations, Dirac equations
Secondary: 35S30: Fourier integral operators 81Q20: Semiclassical techniques, including WKB and Maslov methods

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

Citation

Mizutani, Haruya. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials. Anal. PDE 6 (2013), no. 8, 1857--1898. doi:10.2140/apde.2013.6.1857. https://projecteuclid.org/euclid.apde/1513731448


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