Wave Operators for Schrödinger Operators with Threshold Singuralities, Revisited

Abstract

The continuity property in the Sobolev space $W^{k,p}({\mathbf{R}}^m)$ of wave operators of scattering theory for $m$-dimensional single-body Schrödinger 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 $1<p<m/2$ but not for $p>m/2$ if $m\ge 5$. This extends the previously known interval of $p$ for the continuity, $3/2<p<3$ for $m=3$ and $m/(m-2)<p<m/2$ for $m\ge 5$. The formula which represents the integral kernel of the resolvent of the even dimensional free Schödinger operator as the superposition of exponential-polynomial like functions substantially simplifies the proof of the previous paper when $m\ge 6$ is even.

References in corpus (1)

• The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities

Cited by in corpus (5)

• The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities
• The $L^p$-continuity of wave operators for higher order Schrödinger operators
• On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions
• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators
• Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities