The $L^p$-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions

Abstract

This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Schödinger operator $H={\Delta }^2-\Delta +V(x)$ on ${\mathbb{R}}^n$. Assuming the real-valued potential $V$ exhibits sufficient decay and regularity, we prove that for all dimensions $n\ge 5$, the wave operators $W_{\pm }(H,H_0)$ are bounded on $L^p({\mathbb{R}}^n)$ for all $1\le p\le \infty$, provided that zero is a regular threshold of $H$. As applications, we derive the sharp $L^p$-$L^{p^{\prime }}$ dispersive estimates for Schrödinger group $e^{-itH}$, as well as for the solutions operators $\cos (t\sqrt{H})$ and $\frac{\sin (t\sqrt{H})}{\sqrt{H}}$ associated with the following beam equations with potentials: $${\partial }_t^{2}u+\left({\Delta }^2-\Delta +V(x)\right)u=0,u(0,x)=f(x),{\partial }_{t}u(0,x)=g(x),(t,x)\in \mathbb{R}\times {\mathbb{R}}^n,n\ge 5,$$ where $p^{\prime }$ denotes the Hölder conjugate of $p$, with $1\le p\le 2$. Moreover, we remark that the same results hold for the operator $\epsilon {\Delta }^2-\Delta +V$ with a parameter $\epsilon >0,$ providing greater flexibility for the analysis of related equations.

45 pages