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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2504.05635 |
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| _version_ | 1866909571276079104 |
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| author | Wan, Zijun Yao, Xiaohua |
| author_facet | Wan, Zijun Yao, Xiaohua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $L^p$-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions Wan, Zijun Yao, Xiaohua Analysis of PDEs 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. |
| title | The $L^p$-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.05635 |