Illinois Journal of Mathematics

Dispersive estimates for matrix and scalar Schrödinger operators in dimension five

William R. Green

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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 $\pm\mu$.

Article information

Source
Illinois J. Math. Volume 56, Number 2 (2012), 307-341.

Dates
First available in Project Euclid: 22 November 2013

Permanent link to this document
http://projecteuclid.org/euclid.ijm/1385129950

Mathematical Reviews number (MathSciNet)
MR3161326

Subjects
Primary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Citation

Green, William R. Dispersive estimates for matrix and scalar Schrödinger operators in dimension five. Illinois J. Math. 56 (2012), no. 2, 307--341. http://projecteuclid.org/euclid.ijm/1385129950.


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