Decay estimates for fourth-order Schr\"odinger operators in dimension two

TL;DR

This paper investigates decay estimates for the fourth-order Schrödinger operator in two dimensions, deriving resolvent asymptotics near zero energy and establishing decay rates using harmonic analysis techniques.

Contribution

It provides new asymptotic expansions of the resolvent and classifies zero resonances for the fourth-order Schrödinger operator in 2D, addressing complexities not present in second-order cases.

Findings

Derived resolvent asymptotics near zero energy with resonances or eigenvalues.

Established $L^1-L^$ decay estimates for the evolution operator.

Classified zero resonances as distributional solutions in weighted spaces.

Abstract

In this paper we study the decay estimates of the fourth order Schr\"{o}dinger 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 Schr\"{o}dinger 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 $H\phi=0$ 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 Schr\"odinger operator $-\Delta+V$ in dimension two.