On the 𝐿^{𝑝} boundedness of the wave operators for fourth order Schrödinger operators

Abstract

We consider the fourth order Schrödinger operator H = Δ 2 + V ( x ) H=\Delta ^2+V(x) in three dimensions with real-valued potential V V . Let H 0 = Δ 2 H_0=\Delta ^2 , if V V decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of H H then the wave operators W ± = s \,– lim t → ± ∞ e i t H e − i t H 0 W_{\pm }= s\text {\,–}\lim _{t\to \pm \infty } e^{itH}e^{-itH_0} extend to bounded operators on L p ( R 3 ) L^p(\mathbb R^3) for all 1 > p > ∞ 1>p>\infty .