Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911169698070528 |
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| author | Erdogan, M. Burak Goldberg, Michael Green, William R. |
| author_facet | Erdogan, M. Burak Goldberg, Michael Green, William R. |
| contents | We prove a family of dispersive estimates for the higher order Schrödinger equation $iu_t=(-Δ)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m<n<4m$. Here $V$ is a real-valued potential belonging to the closure of $C_0$ functions with respect to the generalized Kato norm, which has critical scaling. Under standard assumptions on the spectrum, we show that $e^{-itH}P_{ac}(H)$ satisfies a $|t|^{-\frac{n}{2m}}$ bound mapping $L^1$ to $L^\infty$ by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator $(-Δ)^m +V$ and a family of dispersive estimates for operators of the form $|H|^{β-\frac{n}{2m}}e^{-itH}P_{ac}(H)$ for $0<β\leq \frac{n}{2}$. The results apply in both even and odd dimensions in the allowed range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11745 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials Erdogan, M. Burak Goldberg, Michael Green, William R. Analysis of PDEs We prove a family of dispersive estimates for the higher order Schrödinger equation $iu_t=(-Δ)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m<n<4m$. Here $V$ is a real-valued potential belonging to the closure of $C_0$ functions with respect to the generalized Kato norm, which has critical scaling. Under standard assumptions on the spectrum, we show that $e^{-itH}P_{ac}(H)$ satisfies a $|t|^{-\frac{n}{2m}}$ bound mapping $L^1$ to $L^\infty$ by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator $(-Δ)^m +V$ and a family of dispersive estimates for operators of the form $|H|^{β-\frac{n}{2m}}e^{-itH}P_{ac}(H)$ for $0<β\leq \frac{n}{2}$. The results apply in both even and odd dimensions in the allowed range. |
| title | Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2308.11745 |