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具有阈值本征值II的高维Schr dinger算子的色散估计：偶数维情形

Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues II: The even dimensional case

Michael Goldberg, William R. Green

备注：
This article continues the work of "Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues I: The odd dimensional case" by the authors to the case of even dimensions. To appear in J. Spectr. Theory
发表信息：
J. Spectr. Theory, 7 (2017), no. 1, 33-86
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摘要：

我们研究了在偶数维$n\geq 6$中存在零能量本征值时薛定谔算子$H=-Δ+V$的$L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$色散估计.特别是，我们表明，如果有一个本征值在零能量，然后有一个时间依赖的，秩一算子$F_t$满足$\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac{n}{2}}$的$|t|>1$，使$$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\,\,\,\,\,\text{ for } |t|>1.$$与更强的衰减条件的潜力，它是可能产生的运营商值的扩展的演变，采用\begin{align*} e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, \end{align*}的形式，其中$A_{-2}$和$A_{-1}$将$L^1(\mathbb R^n)$映射到$L^\infty(\mathbb R^n)$，而$A_0$将加权的$L^1$空间映射到加权的$L^\infty$空间。首阶项$A_{-2}$和$A_{-1}$都是有限秩的，并且当势能$V$和零能量本征函数之间满足一定的正交条件时，首阶项$A_{-2}$和$A_{-1}$消失。我们表明，在相同的正交条件下，剩余的$|t|^{-\frac{n}{2}}A_0$项也存在作为从$L^1(\mathbb R^n)$到$L^\infty(\mathbb R^n)$的映射，因此$e^{itH}P_{ac}(H)$满足与自由演化相同的色散边界，尽管本征值为零。

英文摘要：

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac{n}{2}}$ for $|t|>1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\,\,\,\,\,\text{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form \begin{align*} e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, \end{align*} with $A_{-2}$ and $A_{-1}$ mapping $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$ while $A_0$ maps weighted $L^1$ spaces to weighted $L^\infty$ spaces. The leading-order terms $A_{-2}$ and $A_{-1}$ are both finite rank, and vanish when certain orthogonality conditions between the potential $V$ and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining $|t|^{-\frac{n}{2}}A_0$ term also exists as a map from $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$, hence $e^{itH}P_{ac}(H)$ satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

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