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Summer 2012 Dispersive estimates for matrix and scalar Schrödinger operators in dimension five
William R. Green
Illinois J. Math. 56(2): 307-341 (Summer 2012). DOI: 10.1215/ijm/1385129950

Abstract

We investigate the boundedness of the evolution operators $e^{itH}$ and $e^{it\mathcal{H}}$ in the sense of $L^{1}\to L^{\infty}$ for both the scalar Schrödinger operator $H=-\Delta+V$ and the non-selfadjoint matrix Schrödinger operator

\begin{eqnarray*}\mathcal{H}=\left [\begin{array}{c@{\quad}c}-\Delta+\mu-V_{1}&-V_{2}\\V_{2}&\Delta-\mu+V_{1}\end{array}\right ]\end{eqnarray*}

in dimension five. Here $\mu>0$ and $V_{1}$, $V_{2}$ are real-valued decaying potentials. The matrix operator arises when linearizing about a standing wave in certain nonlinear partial differential equations. We apply some natural spectral assumptions on $\mathcal{H}$, including regularity of the edges of the spectrum $\pm\mu$.

Citation

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William R. Green. "Dispersive estimates for matrix and scalar Schrödinger operators in dimension five." Illinois J. Math. 56 (2) 307 - 341, Summer 2012. https://doi.org/10.1215/ijm/1385129950

Information

Published: Summer 2012
First available in Project Euclid: 22 November 2013

zbMATH: 1373.35266
MathSciNet: MR3161326
Digital Object Identifier: 10.1215/ijm/1385129950

Subjects:
Primary: 35Q41 , 42B20

Rights: Copyright © 2012 University of Illinois at Urbana-Champaign

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Vol.56 • No. 2 • Summer 2012
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