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

Abstract: We study the $L<sup>p$-boundedness of Isozaki--Kitada type modified wave operators associated with Schrödinger operators $P=-\partial_x² + V$ in one dimension for potentials $V$ including long-range potentials such as negative Coulomb-like ones $V(x) = -(1+|x|<sup>2)<sup>{-μ/2}$ for $μ&gt;0$. For such potentials, if $μ\in (1, 2)$, 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 $μ\in (0, 1]$, we show that even the middle energy part is unbounded except for $p=2$. Actually, regarding the $L<sup>p$-boundedness in the middle and the high energy regimes, we give a complete classification of slowly decaying potentials without imposing any sign condition.