On the observability of the Schrödinger equation in the torus from open sets
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909021221421056 |
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| author | Balc'h, Kévin Le Yu, Jiaqi |
| author_facet | Balc'h, Kévin Le Yu, Jiaqi |
| contents | We study the observability of the Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $ω\subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and for small observation sets of the form $ω= \prod_{j=1}^{d}(a_j,b_j)$. Our second main result shows that observability holds for the Schrödinger equation with a merely bounded potential $V \in L^{\infty}(\mathbb T^d)$, in any dimension $d \geq 1$, for every time $T>0$ and every nonempty open subset $ω$. This resolves a well-known conjecture in the field. A central ingredient in the proof is a cluster decomposition method combined with an induction scheme introduced by Bourgain and further developed by Burq and Zhu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02480 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the observability of the Schrödinger equation in the torus from open sets Balc'h, Kévin Le Yu, Jiaqi Analysis of PDEs Optimization and Control 35Q41, 93B07 We study the observability of the Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $ω\subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and for small observation sets of the form $ω= \prod_{j=1}^{d}(a_j,b_j)$. Our second main result shows that observability holds for the Schrödinger equation with a merely bounded potential $V \in L^{\infty}(\mathbb T^d)$, in any dimension $d \geq 1$, for every time $T>0$ and every nonempty open subset $ω$. This resolves a well-known conjecture in the field. A central ingredient in the proof is a cluster decomposition method combined with an induction scheme introduced by Bourgain and further developed by Burq and Zhu. |
| title | On the observability of the Schrödinger equation in the torus from open sets |
| topic | Analysis of PDEs Optimization and Control 35Q41, 93B07 |
| url | https://arxiv.org/abs/2605.02480 |