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arXiv 1703.00295math.AP
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四阶Schr dinger算子的衰减估计和Schr hartz估计

Decay Estimates and Strichartz Estimates of Fourth-order Schrödinger Operator

Hongliang Feng, Avy Soffer, Xiaohua Yao

备注：
43 Pages, published version. To appear in J. Functional Analysis
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摘要：

研究了四阶薛定谔算子$H=(-Δ)^{2}+V(x)$在$\mathbb{R}^{d}$中对$d=3$和$d\geq5$的时间衰减估计.分析了预解式$R(H; z)$的低能和高能行为，在适当的$H$谱假设下，得到了$e^{-itH}P_{ac}$的Jensen-Kato色散衰减估计和局域衰减估计.基于Jensen-Kato衰减估计和局部衰减估计，利用Ginibre变元得到了$e^{-itH}P_{ac}$在$3$ -维上的$L^1\rightarrow L^{\infty}$估计，并建立了$d\geq5$的$e^{-itH}P_{ac}$在$3$-维上的端点全局估计.进一步，利用局部衰减估计和Georgescu-Larenas-Soffer共轭算子方法，证明了$H$中某些函数的Jensen-Kato型衰减估计.

英文摘要：

We study time decay estimates of the fourth-order Schrödinger operator $H=(-Δ)^{2}+V(x)$ in $\mathbb{R}^{d}$ for $d=3$ and $d\geq5$. We analyze the low energy and high energy behaviour of resolvent $R(H; z)$, and then derive the Jensen-Kato dispersion decay estimate and local decay estimate for $e^{-itH}P_{ac}$ under suitable spectrum assumptions of $H$. Based on Jensen-Kato decay estimate and local decay estimate, we obtain the $L^1\rightarrow L^{\infty}$ estimate of $e^{-itH}P_{ac}$ in $3$-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of $e^{-itH}P_{ac}$ for $d\geq5$. Furthermore, using the local decay estimate and the Georgescu-Larenas-Soffer conjugate operator method, we prove the Jensen-Kato type decay estimates for some functions of $H$.

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