A resonance gap for arbitrary uniformly hyperbolic trapping in three dimensions
Let $V \in C_c^\infty(\mathbb{R}^3; \mathbb{R})$ and $P_h = -h^2 \Delta + V$, and let $E > 0$ be a regular value of $p(x, \xi) = |\xi|^2 + V(x)$, with Hamiltonian flow $\Phi^t$. Let $$K_E = \bigl\{\rho \in p^{-1}(E) : \{\Phi^t(\rho) : t \in \mathbb{R}\} \text{ is bounded}\bigr\}$$ be the trapped set at energy $E$, and assume that it is compact and uniformly hyperbolic: over $K_E$ the tangent bundle of $p^{-1}(E)$ splits continuously and invariantly into the flow direction, a stable subbundle on which $|d\Phi^t v| \le C e^{-ct} |v|$ and an unstable subbundle on which $|d\Phi^{-t} v| \le C e^{-ct} |v|$, for $t \ge 0$. The resonances of $P_h$ are the poles of the meromorphic continuation of the cut-off outgoing resolvent $\chi (P_h - z)^{-1} \chi$, $\chi \in C_c^\infty(\mathbb{R}^3)$, from $\operatorname{Im} z > 0$ across the positive real axis. Nonnenmacher and Zworski proved a resonance-free strip of width proportional to $h$ near $E$, in every dimension, when the topological pressure of the unstable Jacobian on $K_E$ is negative at $1/2$. In dimension two, for potentials whose trapped set is hyperbolic and topologically one-dimensional, Vacossin proved a resonance-free region $\{|z - E| < Ch,\ -\delta h \le \operatorname{Im} z \le 0\}$ with no condition on the pressure.
Problem. Must there exist $\delta, \gamma, h_0 > 0$ such that $P_h$ has no resonance in $$\bigl\{z : |\operatorname{Re} z - E| < \delta,\ -\gamma h < \operatorname{Im} z \le 0\bigr\}$$ for $0 < h < h_0$?
Source: Zworski (2017), Conjecture 3, for operators with a compact hyperbolic trapped set on manifolds with an effective meromorphic continuation; stated by AIM — Open Applied Problems for Schrödinger operators with compactly supported potentials on ℝ³.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, hyperbolic trapped sets, scattering resonances, semiclassical analysis
References
@article{zworski2017,
author = {Zworski, Maciej},
title = {Mathematical study of scattering resonances},
journal = {Bull. Math. Sci.},
volume = {7},
number = {1},
year = {2017},
pages = {1--85},
doi = {10.1007/s13373-017-0099-4},
eprint = {1609.03550},
archivePrefix = {arXiv},
}
@article{nonnenmacher2009,
author = {Nonnenmacher, St{\'e}phane and Zworski, Maciej},
title = {Quantum decay rates in chaotic scattering},
journal = {Acta Math.},
volume = {203},
number = {2},
year = {2009},
pages = {149--233},
doi = {10.1007/s11511-009-0041-z},
eprint = {0706.3242},
archivePrefix = {arXiv},
}
@article{vacossin2024,
author = {Vacossin, Lucas},
title = {Spectral gap for obstacle scattering in dimension 2},
journal = {Anal. PDE},
volume = {17},
number = {3},
year = {2024},
pages = {1019--1126},
doi = {10.2140/apde.2024.17.1019},
eprint = {2201.08259},
archivePrefix = {arXiv},
}
@misc{aim-hyperbolic-trapping-resonance-gap-three-dimensions,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A resonance gap for arbitrary uniformly hyperbolic trapping in three dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/432-hyperbolic-trapping-resonance-gap-three-dimensions.md},
note = {Entry 432 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00443, "A resonance gap for arbitrary uniformly hyperbolic trapping in three dimensions", clopen.solutions.