Decay estimates for fourth-order Schrödinger operators in dimension two

In this paper we study the decay estimates of the fourth order Schrödinger operator H=Δ2+V(x) on R2 with a bounded decaying potential V(x). We first deduce the asymptotic expansions of resolvent of H near zero threshold in the presence of resonances or eigenvalue, and then use them to establish the L1−L∞ decay estimates of e−itH generated by the fourth order Schrödinger operator H. Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we also classify these zero resonances as the distributional solutions of Hϕ=0 in suitable weighted spaces. Due to the degeneracy of Δ2 at zero threshold, we remark that the asymptotic expansions of resolvent RV(λ4) and the classifications of resonances are much more involved than Schrödinger operator −Δ+V in dimension two.