The $L^p$-boundedness of wave operators for fourth order Schrödinger operators on ${\mathbb{R}}^4$

We prove that the wave operators of scattering theory for the fourth order Schrödinger operators ${\mathrm{\Delta }}^2+V(x)$ in ${\mathbb{R}}^4$ are bounded in $L^p({\mathbb{R}}^4)$ for the set of $p$’s of $(1,\mathrm{\infty })$ depending on the kind of spectral singularities of $H$ at zero which can be described by the space of bounded solutions of $({\mathrm{\Delta }}^2+V(x))u(x)=0$.