# The $L_{p}$-boundedness of wave operators for four-dimensional Schrödinger operators

### Kenji Yajima

Gakushuin University, Tokyo, Japan

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## Abstract

We prove that the low energy parts of the wave operators $W_{±}$ for Schrödinger operators $H=−Δ+V(x)$ on $R_{4}$ are bounded in $L_{p}(R_{4})$ for $1<p≤2$ and are unbounded for $2<p≤∞$ 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 $1≤p<4$ in general and for $1≤p<∞$ if all threshold eigenfunctions $φ$ satisfy $∫_{R_{4}}x_{j}V(x)φ(x)dx=0$ for $1≤j≤4$. We prove in this case that they are unbounded in $L_{p}(R_{4})$ for $4<p<∞$ unless the latter condition is satisfied. It is long known that the high energy parts are bounded in $L_{p}(R_{4})$ for all $1≤p≤∞$ and that the same holds for $W_{±}$ if $H$ has no eigenfunctions nor resonances at the threshold.