On Lp boundedness of wave operators for 4-dimensional Schrödinger operators with threshold singularities

Abstract

Let H=−Δ+V(x) be a Schrödinger operator on L2(R4), H0=−Δ. Assume that |V(x)|+|∇ V(x)|⩽C 〈 x 〉−δ for some δ>8. Let W ± = s − l i m t → ± ∞ e i t H e − i t H 0 be the wave operators. It is known that W± extend to bounded operators in Lp(R4) for all 1⩽p⩽∞, if 0 is neither an eigenvalue nor a resonance of H. We show that if 0 is an eigenvalue, but not a resonance of H, then the W± are still bounded in Lp(R4) for all p such that 4/3<p<4.