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

Abstract

We prove that wave operators of scattering theory for fourth order Schr\"odinger operators $H = \Delta^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle x \rangle^{10+\varepsilon} V(x) \in L^1 (\mathbb{R}^2)$ for an $\varepsilon>0$, $\langle x \rangle=(1+|x|^2)^{\frac12}$, 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 $\langle x \rangle^{-1} u(x) \in L^\infty(\mathbb{R}^2)$ 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 $f(\Delta^2)$.