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 $d\ge 5$. 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 $L^1\to L^{\infty }$ 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 $d\ge 5$. 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 $H$.

43 Pages, published version. To appear in J. Functional Analysis