Local smoothing property and Strichartz inequality for Schrodinger equations with potentials superquadratic at infinity (スペクトル・散乱理論とその周辺 研究集会報告集)
Local smoothing property and Strichartz inequality for Schrodinger equations with potentials superquadratic at infinity (スペクトル・散乱理論とその周辺 研究集会報告集)
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DOI：
10.1016/j.jde.2004.03.027
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发表时间：
2002-04
期刊：
影响因子：
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通讯作者：
谷島 賢二;Guoping Zhang
中科院分区：
文献类型：
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作者：
谷島 賢二;Guoping Zhang
研究了含时薛定谔方程i <$u <$t =−（1/2）△u+V（x）u，u（0）=φ∈L2（Rn）的光滑性质，其中位势满足V（x）=O（|X| m）在无穷远处，m <$2。我们证明了在几乎所有的t∈ R上，解u（t，x）是关于x的1/m倍可微的，并解释了这是由于当λ很大时，能量为λ的经典粒子在任意紧集上的逗留时间小于CTλ−1/m during [0，T].我们还证明了这种势的带导数损失的Schehartz不等式，并给出了它在非线性薛定谔方程中的应用。
We study smoothing properties for time-dependent Schrödinger equations i ∂u ∂t =−(1/2)△u+V(x)u , u(0)=φ∈L2( Rn) , with potentials which satisfy V(x)=O(|x|m) at infinity, m⩾2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all t∈ R , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ−1/mduring [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations.