Dispersive estimates for matrix Schr\"{o}dinger operators in dimension two

摘要: We consider the non-selfadjoint operator [\cH = [{array}{cc} -\Delta + \mu-V_1 & -V_2 V_2 \Delta - \mu V_1 {array}]] where $\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around standing wave. Under natural spectral assumptions we obtain $L^1(\R^2)\times L^1(\R^2)\to L^\infty(\R^2)\times L^\infty(\R^2)$ dispersive decay estimates for evolution $e^{it\cH}P_{ac}$. also following weighted estimate $$ \|w^{-1} e^{it\cH}P_{ac}f\|_{L^\infty(\R^2)\times L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)\times L^1(\R^2)},\,\,\,\,\,\,\,\, |t| >2, with $w(x)=\log^2(2+|x|)$.