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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.11753 |
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| _version_ | 1866912886228516864 |
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| author | Galtbayar, Artbazar Yajima, Kenji |
| author_facet | Galtbayar, Artbazar Yajima, Kenji |
| contents | We prove that wave operators of scattering theory for fourth order Schrödinger operators $H = Δ^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle x \rangle^{10+\varepsilon} V(x) \in L^1 (\mathbb{R}^2)$ for an $\varepsilon>0$, $\langle x \rangle=(1+|x|^2)^{\frac12}$, are bounded in $L^p (\mathbb{R}^2)$ for all $1<p<\infty$ if $H$ is regular at zero in the sense that there are no non-trivial solutions to $(Δ^2 + V(x))u(x)=0$ such that $\langle x \rangle^{-1} u(x) \in L^\infty(\mathbb{R}^2)$ and if positive eigenvalues are absent from $H$. This reduces $L^p$-mapping properties of functions $f(H)$ of $H$ to those of Fourier multipliers $f(Δ^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $L^p$-boundedness of wave operators for 4-th order Schrödinger operators on $\mathbb{R}^2$, I. Regular case Galtbayar, Artbazar Yajima, Kenji Mathematical Physics We prove that wave operators of scattering theory for fourth order Schrödinger operators $H = Δ^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle x \rangle^{10+\varepsilon} V(x) \in L^1 (\mathbb{R}^2)$ for an $\varepsilon>0$, $\langle x \rangle=(1+|x|^2)^{\frac12}$, are bounded in $L^p (\mathbb{R}^2)$ for all $1<p<\infty$ if $H$ is regular at zero in the sense that there are no non-trivial solutions to $(Δ^2 + V(x))u(x)=0$ such that $\langle x \rangle^{-1} u(x) \in L^\infty(\mathbb{R}^2)$ and if positive eigenvalues are absent from $H$. This reduces $L^p$-mapping properties of functions $f(H)$ of $H$ to those of Fourier multipliers $f(Δ^2)$. |
| title | The $L^p$-boundedness of wave operators for 4-th order Schrödinger operators on $\mathbb{R}^2$, I. Regular case |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2504.11753 |