Decay Estimates and Strichartz Estimates of Fourth-order Schrödinger Operator (1703.00295v2)
Abstract: We study time decay estimates of the fourth-order Schr\"{o}dinger 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 $L1\rightarrow L{\infty}$ 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 $H$.