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

This paper is devoted to study the time decay estimates for bi-Schrodinger operators $H=(-\Delta)^{2}+V(x)$ in dimension one with the decay potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold with the presence of resonances or eigenvalus; then characterize these resonance spaces corresponding to type of zero resonances in a suitable weighted $L^2({\mathbf{R}})$. Then use them to establish the sharp $L^1-L^\infty$ decay estimates of Schrodinger groups $e^{-itH}$ generated by bi-Schrodinger operators including zero resonances or eigenvalue. As a consequence, the Strichartz estimates are obtained for the solution of fourth-order Schrodinger equations with potentials for initial data in $L^2({\mathbf{R}})$. In particular, it should be emphasized that the presence of zero resonances or eigenvalue does not change the optimal time decay rate of $e^{-itH}$ in dimension one, except at requiring faster decay rate of the potential.