On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators

Abstract

We consider the fourth order Schrödinger operator $H={\Delta }^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0={\Delta }^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ then the wave operators $W_{\pm }=s-{\lim }_{t\to \pm \infty }{e}^{itH}{e}^{-it{H}_0}$ extend to bounded operators on $L^p({\mathbb{R}}^3)$ for all $1<p<\infty$.

arXiv comment22 pages. Fixed typos and addressed comments from referee report. To appear in the Trans. Amer. Math. Soc

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• The $L^p$-continuity of wave operators for higher order Schrödinger operators
• A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators
• Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schrödinger operators