Title: Decay Estimates and Strichartz Estimates of Fourth-order Schrödinger Operator,
Abstract: We study time decay estimates of the fourth-order Schrödinger operator
$H=(-\Delta)^{2}+V(x)$ in $\mathbb{R}^{d}$ for $d=3$ and $d\geq5$. We analyze
the low energy and high energy behaviour of resolvent $R(H; z)$, and then
derive the Jensen-Kato dispersion decay estimate and local decay estimate for
$e^{-itH}P_{ac}$ under suitable spectrum assumptions of $H$. Based on
Jensen-Kato decay estimate and local decay estimate, we obtain the
$L^1\rightarrow L^{\infty}$ estimate of $e^{-itH}P_{ac}$ in $3$-dimension by
Ginibre argument, and also establish the endpoint global Strichartz estimates
of $e^{-itH}P_{ac}$ for $d\geq5$. Furthermore, using the local decay estimate
and the Georgescu-Larenas-Soffer conjugate operator method, we prove the
Jensen-Kato type decay estimates for some functions of $H$.