Decay estimates for bi-Schr\"odinger operators in dimension one

Abstract

This paper is devoted to study the time decay estimates for bi-Schr\"odinger operators $H=\Delta^{2}+V(x)$ in dimension one with decaying potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold without/with the presence of resonances, and then characterize these resonance spaces corresponding to different types of zero resonance in suitable weighted spaces $L_s^2({\mathbf{R}})$. Next we use them to establish the sharp $L^1-L^\infty$ decay estimates of Schr\"odinger groups $e^{-itH}$ generated by bi-Schr\"odinger operators also with zero resonances. As a consequence, Strichartz estimates are obtained for the solution of fourth-order Schr\"odinger equations with potentials for initial data in $L^2({\mathbf{R}})$. In particular, it should be emphasized that the presence of zero resonances does not change the optimal time decay rate of $e^{-itH}$ in dimension one, except at requiring faster decay rate of the potential.