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Dispersive estimates for inhomogeneous fourth-order $Schr\"{o}$dinger operator in 3D with zero energy obstructions

We study the $L^1-L^\infty$ dispersive estimate of the inhomogeneous fourth-order

Schr\"{o}dinger operator $H=\Delta^{2}-\Delta+V(x)$ with zero energy obstructions in

$\mathbf{R}^{3}$. For the related propagator $e^{-itH}$, we prove that for $0<|t|\leq 1$, then

$e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/4}$-dispersive 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 purely a resonance of $H$, there exists a time dependent operator $F_{t}$

such that $e^{-itH}P_{ac}(H)-F_{t}$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 3) if zero is purely

an eigenvalue or zero is both eigenvalue and resonance of $H$, then there exists a time dependent

operator $G_{t}$ such that $e^{-itH}P_{ac}(H)-G_{t}$ satisfies the $|t|^{-3/2}$-dispersive estimate.

Here $F_{t}$ and $G_{t}$ are of finite rank which satisfy $|t|^{-1/2}$-dispersive estimate.