The $L^p$-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions
Zijun Wan, Xiaohua Yao
TL;DR
This work establishes the $L^p$-boundedness of wave operators for a nonhomogeneous fourth-order Schrödinger operator $H=\Delta^2-\Delta+V$ in dimensions $n\ge5$, under decay and regularity assumptions on the real potential $V$ and assuming zero is a regular threshold with no positive embedded eigenvalues. The authors develop a stationary-Born series framework, decomposing the wave operator into an iterative part $W_J^\pm$ and a remainder $\Omega^\pm$, then further splitting $\Omega^\pm$ into low- and high-energy components $\Omega_L^-$ and $\Omega_H^-$. They provide detailed $L^p$-bounds for each piece: the $J$-th iterates via multilinear multiplier representations, the low-energy part via resolvent expansions near zero with invertibility of a finite-rank perturbation, and the high-energy part by oscillatory integral estimates and integration by parts. As applications, the paper derives sharp dispersive $L^p$-$L^{p'}$ estimates for $e^{-itH}$ and for beam-equation propagators $\cos(t\sqrt{H})$ and $\dfrac{\sin(t\sqrt{H})}{\sqrt{H}}$, with similar results valid under the scaled operator $\varepsilon\Delta^2-\Delta+V$ for any $\varepsilon>0$. The results extend the $L^p$-theory of wave operators to nonhomogeneous higher-order operators, enabling robust time-decay and scattering analysis for related PDEs in high dimensions.
Abstract
This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Schödinger operator $H = Δ^2 - Δ+ V(x)$ on $\mathbb{R}^n$. Assuming the real-valued potential $ V $ exhibits sufficient decay and regularity, we prove that for all dimensions $ n \geq 5 $, the wave operators $ W_{\pm}(H, H_0)$ are bounded on $L^{p}(\mathbb{R}^{n}) $ for all $ 1 \leq p \leq \infty $, provided that zero is a regular threshold of $H $. As applications, we derive the sharp $L^p$-$L^{p'}$ 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: $$ \partial_t^2 u + \left(Δ^2 -Δ+ V(x) \right) u = 0, \ \ u(0, x) = f(x), \quad \partial_t u(0, x) = g(x),\ \ (t, x) \in \mathbb{R} \times \mathbb{R}^n,\ n\geq5, $$ where $p'$ denotes the Hölder conjugate of $p$, with $1 \leq p \leq 2$. Moreover, we remark that the same results hold for the operator $ εΔ^2 - Δ+ V$ with a parameter $ε>0,$ providing greater flexibility for the analysis of related equations.