Modified wave operators are unbounded on $L^p$ for $p\neq 2$

Abstract

We study the $L^p$-boundedness of Isozaki--Kitada type modified wave operators associated with Schr\"odinger operators $P=-\partial_x^2 + V$ in one dimension for potentials $V$ including long-range potentials such as negative Coulomb-like ones $V(x) = -(1+|x|^2)^{-\mu/2}$ for $\mu>0$. For such potentials, if $ \mu \in (1, 2)$, we prove that the middle and the high energy parts are bounded on $L^p (\mathbb{R})$ for any $p \in [1, \infty]$ but the low energy part is unbounded except for $p=2$. Moreover, if $\mu \in (0, 1]$, we show that even the middle energy part is unbounded except for $p=2$. Actually, regarding the $L^p$-boundedness in the middle and the high energy regimes, we give a complete classification of slowly decaying potentials without imposing any sign condition.