Decay estimates for fourth-order Schr\"odinger operators in dimension two

In this paper we study the decay estimates of the fourth order Schr\"{o}dinger operator $H=\Delta^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero threshold in the presence of resonances or eigenvalue, and then use them to establish the $L^1-L^\infty$ decay estimates of $e^{-itH}$ generated by the fourth order Schr\"{o}dinger operator $H$. Finally, we classify these zero resonance functions as the distributional solutions of $H\phi=0$ in suitable weighted spaces. Our methods depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, due to the degeneracy of $\Delta^{2}$ at zero threshold and the lower even dimension ( i.e. $n=2$ ), we remark that the asymptotic expansions of resolvent $R_V(\lambda^4)$ and the classifications of resonances are more involved than Schr\"odinger operator $-\Delta+V$ in dimension two.