Dynamical restriction for Schrödinger equations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910071448928256 |
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| author | Nicola, Fabio |
| author_facet | Nicola, Fabio |
| contents | We prove a dynamical restriction principle, asserting that every restriction estimate satisfied by the Fourier transform in $\mathbb{R}^d$ is also valid for the propagator of certain Schrödinger equations. We consider smooth Hamiltonians with an at most quadratic growth, and also a class of nonsmooth Hamiltonians, encompassing potentials that are Fourier transforms of complex (finite) Borel measures. Roughly speaking, if the initial datum belongs to $L^p(\mathbb{R}^d)$, for $p$ in a suitable range of exponents, the solution $u(t,\cdot)$ (for each fixed $t$, with the exception of certain particular values) can be meaningfully restricted to compact curved submanifolds of $\mathbb{R}^d$. The underlying property responsible for this phenomenon is the boundedness of the propagator $L^p\to(\mathcal{F}L^p)_{\rm loc}$, with $1\leq p\leq2$, which is derived from almost diagonalization and dispersive estimates in function spaces defined in terms of wave packet decompositions in phase space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamical restriction for Schrödinger equations Nicola, Fabio Analysis of PDEs Classical Analysis and ODEs Functional Analysis We prove a dynamical restriction principle, asserting that every restriction estimate satisfied by the Fourier transform in $\mathbb{R}^d$ is also valid for the propagator of certain Schrödinger equations. We consider smooth Hamiltonians with an at most quadratic growth, and also a class of nonsmooth Hamiltonians, encompassing potentials that are Fourier transforms of complex (finite) Borel measures. Roughly speaking, if the initial datum belongs to $L^p(\mathbb{R}^d)$, for $p$ in a suitable range of exponents, the solution $u(t,\cdot)$ (for each fixed $t$, with the exception of certain particular values) can be meaningfully restricted to compact curved submanifolds of $\mathbb{R}^d$. The underlying property responsible for this phenomenon is the boundedness of the propagator $L^p\to(\mathcal{F}L^p)_{\rm loc}$, with $1\leq p\leq2$, which is derived from almost diagonalization and dispersive estimates in function spaces defined in terms of wave packet decompositions in phase space. |
| title | Dynamical restriction for Schrödinger equations |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2505.12527 |