Observability and Semiclassical Control for Schrödinger Equations on Non-compact Hyperbolic Surfaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913006529544192 |
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| author | Fu, Xin Gong, Yulin Wang, Yunlei |
| author_facet | Fu, Xin Gong, Yulin Wang, Yunlei |
| contents | We study the observability of the Schrödinger equation on $X$, a non-compact covering space of a compact hyperbolic surface $M$. Using a generalized Bloch theory, functions on $X$ are identified as sections of flat Hilbert bundles over $M$. We develop a semiclassical analysis framework for such bundles and generalize the result of semiclassical control estimates in [Dyatlov and Jin, Acta Math., 220 (2018), pp. 297-339] to all flat Hilbert bundles over $M$, with uniform constants with respect to the choice of bundle. Furthermore, when the Riemannian cover $X \to M$ is a normal cover with a virtually Abelian deck transformation group $Γ$, we combine the uniform semiclassical control estimates on flat Hilbert bundles with the generalized Bloch theory to derive observability from any $Γ$-periodic open subsets of $X$. We also discuss applications of the uniform semiclassical control estimates in spectral geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Observability and Semiclassical Control for Schrödinger Equations on Non-compact Hyperbolic Surfaces Fu, Xin Gong, Yulin Wang, Yunlei Analysis of PDEs Mathematical Physics Spectral Theory We study the observability of the Schrödinger equation on $X$, a non-compact covering space of a compact hyperbolic surface $M$. Using a generalized Bloch theory, functions on $X$ are identified as sections of flat Hilbert bundles over $M$. We develop a semiclassical analysis framework for such bundles and generalize the result of semiclassical control estimates in [Dyatlov and Jin, Acta Math., 220 (2018), pp. 297-339] to all flat Hilbert bundles over $M$, with uniform constants with respect to the choice of bundle. Furthermore, when the Riemannian cover $X \to M$ is a normal cover with a virtually Abelian deck transformation group $Γ$, we combine the uniform semiclassical control estimates on flat Hilbert bundles with the generalized Bloch theory to derive observability from any $Γ$-periodic open subsets of $X$. We also discuss applications of the uniform semiclassical control estimates in spectral geometry. |
| title | Observability and Semiclassical Control for Schrödinger Equations on Non-compact Hyperbolic Surfaces |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2602.14316 |