Decay estimates for fourth-order Schrödinger operators in dimension two

Abstract

In this paper we study the decay estimates of the fourth order Schrödinger operator $H={\Delta }^2+V(x)$ on ${\mathbb{R}}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero threshold in the presence of resonances or eigenvalue, and then use them to establish the $L^1-L^{\infty }$ 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 classify these zero resonances as the distributional solutions of $H\varphi =0$ in suitable weighted spaces. Due to the degeneracy of ${\Delta }^2$ at zero threshold and the lower even dimension (i.e. $n=2$), we remark that the asymptotic expansions of resolvent $R_V({\lambda }^4)$ and the classifications of resonances are more involved than Schrödinger operator $-\Delta +V$ in dimension two.

62 Pages. This is a final version which was published in JFA 2023

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