Strichartz estimates for Schrödinger operators with a non-smoothmagnetic potential

Michael Goldberg

doi:10.3934/dcds.2011.31.109

Article Contents

2011, Volume 31, Issue 1: 109-118. Doi: 10.3934/dcds.2011.31.109

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Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential

Michael Goldberg^1,

1.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025

Received: December 31, 2009

Revised: February 28, 2011

Published: December 2010

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Abstract

We prove Strichartz estimates for the absolutely continuous evolution of a Schrödinger operator $H = (i\nabla + A)^2 + V$ in $R^n$, $n \ge 3$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\rm div}\,A$ or $|\nabla|^{\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators.

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Mathematics Subject Classification: Primary: 35Q20; Secondary: 35C15, 42A16.

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