Dispersive estimates for inhomogeneous fourth-order Schr\"odinger operator in 3D with zero energy obstructions

We study the $L^1-L^\infty$ dispersive estimate of the inhomogeneous fourth-order Schr\"{o}dinger operator $H=\Delta^{2}-\Delta+V(x)$ with zero energy obstructions in $\mathbf{R}^{3}$. For the related propagator $e^{-itH}$, we prove that for $0<t\leq 1$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/4}$-estimate. For $t>1$, we prove that:\,\, 1) if zero is a regular point of $H$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 2) if zero is a resonance of $H$, there exists a time dependent operator $F_{t}$ such that $e^{-itH}P_{ac}(H)-F_{t}$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 3) if zero is a resonance and~/~or an eigenvalue of $H$, then there exists a time dependent operator $G_{t}$ such that $e^{-itH}P_{ac}(H)-G_{t}$ satisfies the $|t|^{-3/2}$- dispersive estimate. Here $F_{t}$ and $G_{t}$ satisfy $|t|^{-1/2}$-dispersive estimates.