Wave Operators for Schr\"odinger Operators with Threshold Singuralities, Revisited

The continuity property in the Sobolev space $W^{k,p}({\mathbf{R}}^m)$ of wave operators of scattering theory for $m$-dimensional single-body Schr"odinger operator is considered when the resolvent of the operator has singularities at the bottom of the continuous spectrum. It is shown that they are continuous in $W^{k,p}({\mathbf{R}}^m)$, $0\le k\le 2$, for $1<p<3$ but not for $p>3$ if $m=3$ and, for $1<p<m\mathrm{/}2$ but not for $p>m\mathrm{/}2$ if $m\ge 5$. This extends the previously known interval of $p$ for the continuity, $3\mathrm{/}2<p<3$ for $m=3$ and $m\mathrm{/}(m-2)<p<m\mathrm{/}2$ for $m\ge 5$. The formula which represents the integral kernel of the resolvent of the even dimensional free Sch"odinger operator as the superposition of exponential-polynomial like functions substantially simplifies the proof of the previous paper when $m\ge 6$ is even.