Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Dispersive estimates for matrix Schrödinger operators in dimension two

Pages: 4473 - 4495, Volume 33, Issue 10, October 2013      doi:10.3934/dcds.2013.33.4473

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M. Burak Erdoǧan - Department of Mathematics, University of Illinois, Urbana, IL 61801, United States (email)
William R. Green - Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN 47803, United States (email)

Abstract: We consider the non-selfadjoint operator \[ H = \left[\begin{array}{cc} -\Delta + \mu-V_1 & -V_2\\ V_2 & \Delta - \mu + V_1 \end{array} \right] \] where $\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\mathbb{R}^2)\times L^1(\mathbb{R}^2)\to L^\infty(\mathbb{R}^2)\times L^\infty(\mathbb{R}^2)$ dispersive decay estimates for the evolution $e^{it H}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\mathcal H}P_{ac}f\|_{L^\infty(\mathbb R^2)\times L^\infty(\mathbb R^2)} ≲ \frac{1}{|t|\log^2(|t|)} \|w f\|_{L^1(\mathbb R^2)\times L^1(\mathbb R^2)},\,\,\,\,\,\,\,\, |t| >2, $$with $w(x)=\log^2(2+|x|)$.

Keywords:  Matrix Schrödinger operators, dispersive estimates, weighted estimates, solitons, asymptotic stability.
Mathematics Subject Classification:  Primary: 35J10, 35Q40.

Received: November 2012;      Revised: February 2013;      Available Online: April 2013.