Pointwise estimates for the fundamental solutions of higher order schrödinger equations with finite rank perturbations
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| Format: | Preprint |
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2025
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| _version_ | 1866913636703797248 |
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| author | Chen, Xinyi Cheng, Han Huang, Shanlin |
| author_facet | Chen, Xinyi Cheng, Han Huang, Shanlin |
| contents | This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schrödinger equation: % we investigate the fundamental solution of the higher order Schrödinger equation $$i{\partial}_{t}u(x,t)=Hu(x,t),\ \ \ t\in \mathbb{R},\ x\in {\mathbb{R}}^{n},$$ where the Hamiltonian $H$ is defined as $$H={(-Δ)}^{m}+\displaystyle\sum_{j=1}^{N} \langle\cdotp ,{φ}_{j} \rangle{φ}_{j},$$ with each $φ_j$ ($1\le j\le N$) satisfying certain smoothness and decay conditions. %Let ${P}_{ac}(H)$ denote the projection onto the absolutely continuous space of $H$. We show that for any positive integer $m>1$ and spatial dimension $n\ge 1$, %under a spectral assumption, the operator is sharp in the sense that it
${e}^{-i tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the following pointwise estimate: $$\left |K(t,x,y)\right |\lesssim |t|^{-\frac{n}{2m}}(1+|t|^{-\frac{1}{2m}}\left | x-y\right |)^{-\frac{n(m-1)}{2m-1}} ,\ \ t\ne 0,\ x,y\in {\mathbb{R}}^{n}.$$ This estimate is consistent with the upper bounds in the free case. As an application, we derive $L^p-L^q$ decay estimates for the propagator ${e}^{-ıtH}P_{ac}(H)$, where the pairs $(1/p, 1/q)$ lie within a quadrilateral region in the plane. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_02562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointwise estimates for the fundamental solutions of higher order schrödinger equations with finite rank perturbations Chen, Xinyi Cheng, Han Huang, Shanlin Analysis of PDEs Classical Analysis and ODEs This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schrödinger equation: % we investigate the fundamental solution of the higher order Schrödinger equation $$i{\partial}_{t}u(x,t)=Hu(x,t),\ \ \ t\in \mathbb{R},\ x\in {\mathbb{R}}^{n},$$ where the Hamiltonian $H$ is defined as $$H={(-Δ)}^{m}+\displaystyle\sum_{j=1}^{N} \langle\cdotp ,{φ}_{j} \rangle{φ}_{j},$$ with each $φ_j$ ($1\le j\le N$) satisfying certain smoothness and decay conditions. %Let ${P}_{ac}(H)$ denote the projection onto the absolutely continuous space of $H$. We show that for any positive integer $m>1$ and spatial dimension $n\ge 1$, %under a spectral assumption, the operator is sharp in the sense that it ${e}^{-i tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the following pointwise estimate: $$\left |K(t,x,y)\right |\lesssim |t|^{-\frac{n}{2m}}(1+|t|^{-\frac{1}{2m}}\left | x-y\right |)^{-\frac{n(m-1)}{2m-1}} ,\ \ t\ne 0,\ x,y\in {\mathbb{R}}^{n}.$$ This estimate is consistent with the upper bounds in the free case. As an application, we derive $L^p-L^q$ decay estimates for the propagator ${e}^{-ıtH}P_{ac}(H)$, where the pairs $(1/p, 1/q)$ lie within a quadrilateral region in the plane. |
| title | Pointwise estimates for the fundamental solutions of higher order schrödinger equations with finite rank perturbations |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2501.02562 |