The $L^p$-boundedness of wave operators for 4-th order Schrödinger operators on ${\mathbb{R}}^2$, I. Regular case

Abstract

We prove that wave operators of scattering theory for fourth order Schrödinger operators $H={\Delta }^2+V(x)$ on ${\mathbb{R}}^2$ with real potentials $V(x)$ such that $⟨x{⟩}^{3}V(x)\in L^{\frac{4}{3}}({\mathbb{R}}^2)$ and $⟨x{⟩}^{10+\epsilon }V(x)\in L^1({\mathbb{R}}^2)$ for an $\epsilon >0$, $⟨x⟩=(1+|x{|}^2{)}^{\frac{1}{2}}$, are bounded in $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$ if $H$ is regular at zero in the sense that there are no non-trivial solutions to $({\Delta }^2+V(x))u(x)=0$ such that $⟨x{⟩}^{-1}u(x)\in L^{\infty }({\mathbb{R}}^2)$ and if positive eigenvalues are absent from $H$. This reduces $L^p$-mapping properties of functions $f(H)$ of $H$ to those of Fourier multipliers $f({\Delta }^2)$.