Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in odd dimensions II: high dimensional case
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| Format: | Preprint |
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2024
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| _version_ | 1866917795051077632 |
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| author | Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan |
| author_facet | Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan |
| contents | In this paper, for any odd $n$ and any integer $m\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schrödinger equation \begin{equation*} \mathrm{i}\partial_tu(x,t)=((-Δ)^m+V(x))u(x,t),\quad t\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is a real-valued $C^{\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-Δ)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K(t,x,y)$ satisfying \begin{equation*} |K(t, x,y)|\le C(1+|t|)^{-(\frac{n}{2m}-σ)}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, \end{equation*} where $σ=2$ if $0$ is an eigenvalue of $H$, and $σ=0$ otherwise. A similar result for smoothing operators $H^\fracα{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. The regularity condition $V\in C^{\frac{n+1}{2}-2m}$ is optimal in the second order case, and it also seems optimal when $m>1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in odd dimensions II: high dimensional case Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan Analysis of PDEs In this paper, for any odd $n$ and any integer $m\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schrödinger equation \begin{equation*} \mathrm{i}\partial_tu(x,t)=((-Δ)^m+V(x))u(x,t),\quad t\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is a real-valued $C^{\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-Δ)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K(t,x,y)$ satisfying \begin{equation*} |K(t, x,y)|\le C(1+|t|)^{-(\frac{n}{2m}-σ)}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, \end{equation*} where $σ=2$ if $0$ is an eigenvalue of $H$, and $σ=0$ otherwise. A similar result for smoothing operators $H^\fracα{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. The regularity condition $V\in C^{\frac{n+1}{2}-2m}$ is optimal in the second order case, and it also seems optimal when $m>1$. |
| title | Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in odd dimensions II: high dimensional case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.00117 |