$L^p$ boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues
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arXiv
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2025
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| _version_ | 1866913903686975488 |
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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 $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when $n>4m$ to lower dimensions $2m<n\leq 4m$ to show that when $H$ has a threshold eigenvalue and no resonances, the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{2n}{n-1}$ when $n$ is odd and $1\leq p<\frac{2n}{n-2}$ when $n$ is even. We further show that if the zero energy eigenfunctions are orthogonal to $x^αV(x)$ for all $|α|<k_0$, then the wave operators are bounded on $1\leq p<\frac{n}{2m-k_0}$ when $k_0<2m$ in all dimensions $n>2m$. The range is $p\in [1,\infty)$ and $p\in[1,\infty]$ when $k_0=2m$ and $k_0>2m$ respectively. The proofs apply in the classical $m=1$ case as well and streamlines existing arguments in the eigenvalue only case, in particular the $L^\infty(\mathbb R^n)$ boundedness is new when $n>3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p$ boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues Erdogan, M. Burak Green, William R. LaMaster, Kevin Analysis of PDEs Spectral Theory We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when $n>4m$ to lower dimensions $2m<n\leq 4m$ to show that when $H$ has a threshold eigenvalue and no resonances, the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{2n}{n-1}$ when $n$ is odd and $1\leq p<\frac{2n}{n-2}$ when $n$ is even. We further show that if the zero energy eigenfunctions are orthogonal to $x^αV(x)$ for all $|α|<k_0$, then the wave operators are bounded on $1\leq p<\frac{n}{2m-k_0}$ when $k_0<2m$ in all dimensions $n>2m$. The range is $p\in [1,\infty)$ and $p\in[1,\infty]$ when $k_0=2m$ and $k_0>2m$ respectively. The proofs apply in the classical $m=1$ case as well and streamlines existing arguments in the eigenvalue only case, in particular the $L^\infty(\mathbb R^n)$ boundedness is new when $n>3$. |
| title | $L^p$ boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2506.16378 |