Boundedness of Wave Operators for Schrödinger Operators with Threshold Singuralities I . The Odd Dimensional Case
Boundedness of Wave Operators for Schrödinger Operators with Threshold Singuralities I . The Odd Dimensional Case
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具有阈值奇点的薛定谔算子的波算子有界性 I. 奇维情况
DOI：
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
K.
中科院分区：
文献类型：
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作者：
K.
设H = − H + V（x）是m维奇薛定谔算子，m ≥ 3，H 0 = − H，W± = lim t→±∞ eitHe− itH 0是（H，H 0）的波算子。如果0不是H的本征值也不是H的共振，我们说H是属型的，否则说H是例外型的。我们假设V满足F（<$x <$−2σV）∈ L <$$>，对于某些σ > 1 m <$，m <$= m−1 m−2。我们证明了，对于所有的1 ≤ p ≤ ∞，如果V另外满足|V（x）|≤ C <$x <$−m−2−ε，对于某个ε > 0，且H是类属型;且对于m m−2和m2之间的所有p，W±在L（R）中有界，但对于闭区间[ m m−2，m 2 ]之外的p，W±在L（R）中有界，如果V满足|V（x）|≤ C <$x <$−m−3−ε且若H是例外型。这特别意味着传播算子的连续部分满足L-L估计，即：|不|1 m（1 2− 1 q <$u <$q，对于对偶指数1 p + 1 q = 1使得1 ≤ q ≤ 2 ≤ p ≤ ∞，如果H是一般类型，且对于m m−2 < q ≤ 2 ≤ p < m2，m ≥ 5，或32 < q ≤ 2 ≤ p < 3，m = 3，如果H是例外类型。
Let H = −∆ + V (x) be an odd m-dimensional Schrödinger operator, m ≥ 3, H0 = −∆, and let W± = lim t→±∞ eitHe−itH0 be the wave operators for the pair (H,H0). We say H is of generic type if 0 is not an eigenvalue nor a resonance of H and of exceptional type if otherwise. We assume that V satisfies F(〈x〉−2σV ) ∈ L∗ for some σ > 1 m∗ , m∗ = m−1 m−2 . We show that W± are bounded in L(R ) for all 1 ≤ p ≤ ∞ if V satisfies in addition |V (x)| ≤ C〈x〉−m−2−ε for some ε > 0 and if H is of generic type; and that W± are bounded in L(R) for all p between m m−2 and m2 but not for p outside the closed interval [ m m−2 , m 2 ] if V satisfies |V (x)| ≤ C〈x〉−m−3−ε and if H is of exceptional type. This in particular implies that the continuous part of the propagator satisfies the L-L estimates ‖e−itHPc(H)u‖p ≤ C|t| 1 m ( 1 2− 1 q ‖u‖q for the dual exponents 1 p + 1 q = 1 such that 1 ≤ q ≤ 2 ≤ p ≤ ∞ if H is of generic type, and for m m−2 < q ≤ 2 ≤ p < m2 , m ≥ 5, or 32 < q ≤ 2 ≤ p < 3, m = 3, if H of exceptional type.
具有时间周期势的薛定谔方程解的局部时间衰减。
DOI：
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发表时间：
2004
期刊：
J.Statist.Phys. 116
影响因子：
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作者：
Galtbayar;A.;Jensen;A.;Yajima;K.
通讯作者：
K.