论文标题
对$ l^p $的反示例，用于古典和高级施罗丁运营商的波浪运营商的界面
Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schrödinger operators
论文作者
论文摘要
我们考虑高阶Schrödinger运算符$ h =（ - δ）^m+v（x）$ in $ n $ dimensions in $ n $ dimensions，当$ n> 4M-1 $，$ m \ in \ mathbb n $时，具有实值的潜在$ v $。我们表明，对于任何$ \ frac {2n} {n-4m+1} <p \ leq \ infty $和$ 0 \ leqleqα<\ frac {n+1} {2} {2} -2m- \ frac {n} p $ Wave Operator $ W^{\ PM} $在$ l^p（\ Mathbb r^n）$上不限制。由于我们的分析，我们表明，通常二阶schrödingeroperator $-Δ+v $的波浪运算符在$ l^p（\ mathbb r^n）$上无限制，$ n> 3 $> 3 $和$ \ frac {2n} {n-3} {n-3} {n-3} <p \ le leq \ le f \ le f \ leq \ lefty nighte nigelsife toffully toxpect y toress $ v $ $ l^{p'} \至l^p $分散估计可能具有独立利益。
We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in \mathbb N$. We show that for any $\frac{2n}{n-4m+1}<p\leq \infty$ and $0\leq α<\frac{n+1}{2}-2m-\frac{n}p$, there exists a real-valued, compactly supported potential $V\in C^α(\mathbb R^n)$ for which the wave operators $W^{\pm}$ are not bounded on $L^p(\mathbb R^n)$. As a consequence of our analysis we show that the wave operators for the usual second order Schrödinger operator $-Δ+V$ are unbounded on $L^p(\mathbb R^n)$ for $n>3$ and $\frac{2n}{n-3}<p\leq \infty$ for insufficiently differentiable potentials $V$, and show a failure of $L^{p'}\to L^p$ dispersive estimates that may be of independent interest.
