Robust Observability for Schrödinger Equations with Singular Potentials on 2D Compact Hyperbolic Surfaces
Abstract
This paper investigates the robustness of quantum observability on compact hyperbolic surfaces subject to Lq potential perturbations with 2<q≤ ∞. Since Lq regularity is strictly subcritical in dimension two, for every non-empty open set Ω⊂ M and T>0, the solution of (i∂t+Δ−V)u=0 satisfies the space-time observability estimate |u0|L2(M)2 ≤ C∫0T |e−it(−Δ+V)u0|L2(Ω)2 dt. A cornerstone of our analysis is Sogge's spectral cluster estimate, which allows us to obtain refined spectrally localized semiclassical Strichartz estimates adapted to rough potentials and to show that the potential's contribution vanishes in the propagation of semiclassical measures. The full-support property of invariant semiclassical measures on hyperbolic surfaces associated with the eigenfunctions then yields the desired observability by contradiction. By the Hilbert Uniqueness Method, the corresponding internal controllability result follows.
Related
- Observability for wave equations with critically singular potentials
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
- Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials
- Observability Inequalities for Degenerate Schrödinger Equations via a Shape Approximation Method
- Semigroup properties of Schrödinger operators with complex singular potentials
BibTeX
@misc{wu2026robust,
title = {{Robust Observability for Schrödinger Equations with Singular Potentials on 2D Compact Hyperbolic Surfaces}},
author = {Jiayan Wu and Ting Zhang and Ruze Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34287},
eprint = {2610.34287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34287},
}
RIS
TY - UNPB
TI - Robust Observability for Schrödinger Equations with Singular Potentials on 2D Compact Hyperbolic Surfaces
AU - Jiayan Wu
AU - Ting Zhang
AU - Ruze Zhou
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34287
UR - https://arxiv.org/abs/2610.34287
AB - This paper investigates the robustness of quantum observability on compact hyperbolic surfaces subject to L^q potential perturbations with 2<q≤ ∞. Since L^q regularity is strictly subcritical in dimension two, for every non-empty open set Ω⊂ M and T>0, the solution of (i∂_t+Δ−V)u=0 satisfies the space-time observability estimate |u_0|_L^2(M)^2 ≤ C∫_0^T |e^−it(−Δ+V)u_0|_L^2(Ω)^2 dt. A cornerstone of our analysis is Sogge's spectral cluster estimate, which allows us to obtain refined spectrally localized semiclassical Strichartz estimates adapted to rough potentials and to show that the potential's contribution vanishes in the propagation of semiclassical measures. The full-support property of invariant semiclassical measures on hyperbolic surfaces associated with the eigenfunctions then yields the desired observability by contradiction. By the Hilbert Uniqueness Method, the corresponding internal controllability result follows.
ER -