论文标题
具有奇异电位的时间依赖性schrödinger方程的Quasimode和Strichartz估计值
Quasimode and Strichartz estimates for time-dependent Schrödinger equations with singular potentials
论文作者
论文摘要
我们通过允许巨大的奇异潜力$ v $概括了Schrödinger经营者的Strichartz估计值。具体而言，我们表明他们的$ 1/p $ - 损失$ l^p_tl^q_x（i \ times m）$ - strichartz估计估计$ e^{ - ith_v} $时，$ e^{ - h_v =-Δ_g+v（x）$ with $ v \ in l^{n/2}（m） l^{1+δ}（m）$，$δ> 0 $，如果$ n = 2 $，则$（p，q）$如龙骨-TAO定理和$ i \ subset {\ subset {\ mathbb r} $ a有界间隔。我们通过为缩放的二元二元型不受干扰的Schrödinger操作员制定和证明新的“ Quasimode”估计值，并利用$ 1/Q'-1/q = 2/n $的端点strichartz估计$（p，q）=（2,2n/（n-2））$的事实。我们还表明，在{\ em Any}紧凑型歧管上获得的通用准莫德估计值饱和。但是，我们建议他们可以使用最近开发的“ Kakeya-Nikodym”技术来改善某些几何形状的Strichartz估计，从而获得改进的特征功能估计值，例如，假设为负曲率。
We generalize the Strichartz estimates for Schrödinger operators on compact manifolds of Burq, Gérard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times M)$-Strichartz estimates hold for $e^{-itH_V}$ when $H_V=-Δ_g+V(x)$ with $V\in L^{n/2}(M)$ if $n\ge3$ or $V\in L^{1+δ}(M)$, $δ>0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao theorem and $I\subset {\mathbb R}$ a bounded interval. We do this by formulating and proving new "quasimode" estimates for scaled dyadic unperturbed Schrödinger operators and taking advantage of the the fact that $1/q'-1/q=2/n$ for the endpoint Strichartz estimates when $(p,q)=(2,2n/(n-2))$. We also show that the universal quasimode estimates that we obtain are saturated on {\em any} compact manifolds; however, we suggest that they may lend themselves to improved Strichartz estimates in certain geometries using recently developed "Kakeya-Nikodym" techniques developed to obtain improved eigenfunction estimates assuming, say, negative curvatures.
