论文标题
Wave操作员的$ l^p $结合度具有阈值共振的四维Schrödinger操作员
The $L^p$-boundedness of wave operators for four dimensional Schrödinger operators with threshold resonances
论文作者
论文摘要
我们证明，schrödinger运算符的Wave运算符的低能部分$ w_ \ pm $ $ h = - \ lap + v（x）$ on $ \ r^4 $在$ l^p（\ r^4）$中以$ 1 <p \ leq 2 $ 2 $中的$ l^p（\ r^4）$限制，并以$ 2 <p \ leq \ leq \ effty $ reson reson reson reson reson bith und of。如果$ h $仅在阈值时具有本本函数，那么最近证明，它们在$ l^p（\ r^4）$中以$ 1 \ leq p <4 $一般的限制，如果$ 1 \ leq p <\ f \ \ infty $，如果全部阈值eigenfunctions $ \ \ ph $ \ ph $ \ ph $ \ ph $ \ int in Int _^\ r^4 \ pH（x）dx = 0 $ for $ 1 \ leq j \ leq 4 $。在这种情况下，我们证明它们在$ l^p（\ r^4）$中以$ 4 <p <\ infty $无限，除非满足后一种条件。众所周知，对于所有$ 1 \ leq p \ leq \ leq \ infty $，高能零件在$ l^p（\ r^4）$中限制，如果$ h $没有特征性或共鸣。
We prove that the low energy parts of the wave operators $W_\pm$ for Schrödinger operators $H = -\lap + V(x)$ on $\R^4$ are bounded in $ L^p(\R^4)$ for $1<p\leq 2$ and are unbounded for $2<p\leq \infty$ if $H$ has resonances at the threshold. If $H$ has eigenfunctions only at the threshold, it has recently been proved that they are bounded in $L^p(\R^4)$ for $1\leq p<4$ in general and for $1\leq p<\infty$ if all threshold eigenfunctions $\ph$ satisfy $\int_{\R^4}x_j V(x) \ph(x)dx=0$ for $1\leq j\leq 4$. We prove in this case that they are unbounded in $L^p(\R^4)$ for $4<p<\infty$ unless the latter condition is satisfied. It is long known that the high energy parts are bounded in $L^p(\R^4)$ for all $1\leq p\leq \infty$ and that the same holds for $W_\pm$ if $H$ has no eigenfunctions nor resonances at the threshold.
