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

Ping Li; Avy Soffer; Xiaohua Yao

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arXiv · 2110.07154

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

Ping Li·Avy Soffer·Xiaohua Yao

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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.

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Ping Li, Avy Soffer, Xiaohua Yao. 2021-10-14. Decay estimates for fourth-order Schr\"odinger operators in dimension two. https://arxiv.org/abs/2110.07154

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