A fractal Weyl lower bound for obstacle resonances
Let $\mathcal{O} \subset \mathbb{R}^2$ be a union of at least three pairwise disjoint closed disks satisfying the no-eclipse condition: the convex hull of any two of the disks is disjoint from every other disk. Let $\mathcal{R}$ be the set of outgoing scattering resonances of the Dirichlet Laplacian on $\mathbb{R}^2 \setminus \mathcal{O}$ in the wavenumber variable $z$: the poles of the meromorphic continuation of $(-\Delta - z^2)^{-1}$ adjacent to the positive real axis, counted with algebraic multiplicity. Let $K$ be the trapped set, the unit-speed billiard states (a point of $\mathbb{R}^2 \setminus \mathcal{O}$ and a unit direction) whose trajectories remain bounded for both positive and negative time, and write $\dim_H K = 2\delta + 1$. Numerical computations of Lu, Sridhar and Zworski for three-disk systems, based on semiclassical zeta functions, suggest that the number of resonances in a strip below the real axis with real part at most $R$ grows like $R^{1+\delta}$. Nonnenmacher, Sjöstrand and Zworski proved the corresponding upper bound for planar strictly convex obstacles satisfying the no-eclipse condition: for fixed $C > 0$ there are $O(r^{\delta})$ resonances with $\operatorname{Im} z > -C$ and $r \le |z| \le r + 1$.
Problem. Is there $C_0 > 0$ such that, for every fixed $C > C_0$, there are $c_C, R_C > 0$ with $$\#\{z \in \mathcal{R} : 1 \le \operatorname{Re} z \le R,\ -C \le \operatorname{Im} z < 0\} \ge c_C R^{1+\delta} \qquad (R \ge R_C)?$$
Source: Lu, Sridhar and Zworski (2003), as a fractal Weyl law for scattering by several disks; see Nonnenmacher, Sjöstrand and Zworski (2014), (1.4), and Zworski (2017), Conjecture 5. Stated by AIM — Open Applied Problems as a lower bound.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, DS
- Keywords
- AIM Open Applied Problems, fractal Weyl laws, obstacle scattering, scattering resonances
References
@article{lu2003,
author = {Lu, W. T. and Sridhar, S. and Zworski, Maciej},
title = {Fractal {W}eyl laws for chaotic open systems},
journal = {Phys. Rev. Lett.},
volume = {91},
number = {15},
year = {2003},
pages = {154101},
doi = {10.1103/PhysRevLett.91.154101},
eprint = {nlin/0208019},
archivePrefix = {arXiv},
}
@article{nonnenmacher2014,
author = {Nonnenmacher, St{\'e}phane and Sj{\"o}strand, Johannes and Zworski, Maciej},
title = {Fractal {W}eyl law for open quantum chaotic maps},
journal = {Ann. of Math. (2)},
volume = {179},
number = {1},
year = {2014},
pages = {179--251},
doi = {10.4007/annals.2014.179.1.3},
eprint = {1105.3128},
archivePrefix = {arXiv},
}
@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},
}
@misc{aim-fractal-weyl-resonances,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A fractal Weyl lower bound for obstacle resonances}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/017-fractal-weyl-resonances.md},
note = {Entry 017 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-00286, "A fractal Weyl lower bound for obstacle resonances", clopen.solutions.