On L p boundedness of wave operators for 4‐dimensional Schrödinger operators with threshold singularities

Let H =−Δ+ V ( x ) be a Schrödinger operator on L 2 ( R 4 ), H 0 =−Δ. Assume that | V ( x )|+|∇ V ( x )|⩽ C 〈 x 〉 −δ for some δ>8. Let W ± = s − l i m t → ± ∞e i t He − i t H 0be the wave operators. It is known that W ± extend to bounded operators in L p ( R 4 ) 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 L p ( R 4 ) for all p such that 4/3< p <4.