Wave Operators for Schrödinger Operators with Threshold Singuralities, Revisited

الأصل · EN

The continuity property in the Sobolev space Wᵏ,ᵖ(Rᵐ) 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 Wᵏ,ᵖ(Rᵐ), 0≤ k ≤ 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≥ 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≥ 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 ≥ 6 is even.