$L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912268951748608 |
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| author | Erdogan, M. Burak Green, William R. LaMaster, Kevin |
| author_facet | Erdogan, M. Burak Green, William R. LaMaster, Kevin |
| contents | We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that when $H$ has a threshold eigenvalue the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{n}{2m}$ in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when $m=1$. The proof applies in the classical $m=1$ case as well and simplifies the argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions Erdogan, M. Burak Green, William R. LaMaster, Kevin Analysis of PDEs Mathematical Physics We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that when $H$ has a threshold eigenvalue the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{n}{2m}$ in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when $m=1$. The proof applies in the classical $m=1$ case as well and simplifies the argument. |
| title | $L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2407.07069 |