Infinitely many obstacle resonances in a fixed strip under trapping
Let $K \subset \mathbb{R}^3$ be the closure of a bounded open set with smooth boundary whose exterior $\Omega = \mathbb{R}^3 \setminus K$ is connected, and let $-\Delta_D$ be the Dirichlet Laplacian on $\Omega$. Its scattering resonances $\operatorname{Res}(K)$ are the poles of the meromorphic continuation of the outgoing resolvent $$(-\Delta_D - z^2)^{-1} : L^2_{\mathrm{comp}}(\Omega) \to H^2_{\mathrm{loc}}(\Omega)$$ from $\operatorname{Im} z > 0$, with the outgoing convention $e^{izr}/r$, so that resonances lie in $\operatorname{Im} z < 0$. The obstacle is trapping if some ray of geometric optics, reflected specularly at the boundary (as a generalized ray at tangencies), stays in a bounded region for all forward time. For nontrapping obstacles every strip $\{-C \le \operatorname{Im} z < 0\}$ contains only finitely many resonances. Lax and Phillips conjectured that a trapping obstacle has resonances with imaginary parts tending to zero; Ikawa showed that two disjoint strictly convex obstacles have a resonance-free strip below the real axis, and proposed the weaker statement below. It holds for two disjoint strictly convex obstacles (Ikawa; Gérard), for unions of sufficiently small balls no three of whose centres are collinear (Ikawa), and for unions of at least three disjoint strictly convex obstacles with real-analytic boundary satisfying the non-eclipse condition (Chaubet and Petkov).
Problem. Is it true that for every trapping obstacle $K$ as above there is $C = C(K) > 0$ such that $$\#\{z \in \operatorname{Res}(K) : -C \le \operatorname{Im} z < 0\} = \infty?$$ The resonances need not approach the real axis.
Source: Ikawa (1990), the modified Lax–Phillips conjecture, after Lax and Phillips (1967); in this form, with resonances in the lower half-plane, Alves and Antunes (2024), Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS, SP
- Keywords
- AIM Open Applied Problems, billiards, obstacle scattering, scattering resonances
References
@inproceedings{ikawa1990jedp,
author = {Ikawa, Mitsuru},
title = {On poles of scattering matrices for several convex bodies},
booktitle = {Journ{\'e}es {\'E}quations aux d{\'e}riv{\'e}es partielles},
year = {1990},
pages = {Exp. No. VI, 1--9},
doi = {10.5802/jedp.388},
}
@incollection{ikawa1990lnm,
author = {Ikawa, Mitsuru},
title = {On the distribution of poles of the scattering matrix for several convex bodies},
booktitle = {Functional-Analytic Methods for Partial Differential Equations (Tokyo, 1989)},
series = {Lecture Notes in Math.},
volume = {1450},
publisher = {Springer},
address = {Berlin},
year = {1990},
pages = {210--225},
doi = {10.1007/BFb0084908},
}
@book{laxphillips1989,
author = {Lax, Peter D. and Phillips, Ralph S.},
title = {Scattering Theory},
edition = {Second},
series = {Pure and Applied Mathematics},
volume = {26},
publisher = {Academic Press},
address = {Boston, MA},
year = {1989},
note = {First edition 1967},
}
@article{alves2024,
author = {Alves, Carlos J. S. and Antunes, Pedro R. S.},
title = {Wave scattering problems in exterior domains with the method of fundamental solutions},
journal = {Numer. Math.},
volume = {156},
number = {2},
year = {2024},
pages = {375--394},
doi = {10.1007/s00211-024-01395-x},
}
@article{ikawa1983,
author = {Ikawa, Mitsuru},
title = {On the poles of the scattering matrix for two strictly convex obstacles},
journal = {J. Math. Kyoto Univ.},
volume = {23},
number = {1},
year = {1983},
pages = {127--194},
doi = {10.1215/kjm/1250521614},
}
@article{gerard1988,
author = {G{\'e}rard, C.},
title = {Asymptotique des p{\^o}les de la matrice de scattering pour deux obstacles strictement convexes},
journal = {M{\'e}m. Soc. Math. France (N.S.)},
number = {31},
year = {1988},
pages = {1--146},
doi = {10.24033/msmf.332},
}
@article{chaubet2026,
author = {Chaubet, Yann and Petkov, Vesselin},
title = {Dynamical zeta functions for billiards},
journal = {Ann. Inst. Fourier (Grenoble)},
volume = {76},
number = {4},
year = {2026},
pages = {1537--1592},
doi = {10.5802/aif.3743},
eprint = {2201.00683},
archivePrefix = {arXiv},
}
@misc{aim-modified-lax-phillips-trapping,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Infinitely many obstacle resonances in a fixed strip under trapping}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/581-modified-lax-phillips-trapping.md},
note = {Entry 581 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-00363, "Infinitely many obstacle resonances in a fixed strip under trapping", clopen.solutions.