Real-energy nullspace on nontrapping Lorentzian scattering spaces
Let $M$ be a connected compact smooth manifold with boundary, $\dim M \ge 2$, with boundary defining function $x$, and let $g$ be a smooth Lorentzian scattering metric, that is, a Lorentzian metric on the scattering tangent bundle, whose local frame is $x^2\partial_x, x\partial_{y_j}$. Let $G$ be the dual metric function and $\dot{C}^\infty(M)$ the smooth functions vanishing to infinite order at $\partial M$. For real $\lambda \ne 0$, assume the hypotheses of Vasy: the rescaled Hamilton flow of $G$ has submanifolds $L_-$ and $L_+$ at base infinity, transversal to fibre infinity, which are normal sources and normal sinks respectively, and every integral curve outside $L_\pm$, in $\{G = 0\}$ at fibre infinity and in $\{G = \lambda\}$ at base infinity, tends to $L_+$ forward and to $L_-$ backward.
Vasy proved that if $\square_g - \lambda$ has trivial nullspace on the Feynman or the anti-Feynman function space, then the limit of $(\square_g - (\lambda \pm i\epsilon))^{-1}$ as $\epsilon \to 0$ exists in the weak operator topology on that space (sign $+$ for Feynman, $-$ for anti-Feynman) and is the corresponding propagator, and that both nullspaces lie in $\dot{C}^\infty(M)$. In signature $(1, n-1)$ he observes that they are trivial for $\lambda > 0$ on the narrower class of Lorentzian scattering metrics of Baskin, Vasy and Wunsch, whose $L_\pm$ are conormal bundles, having a boundary defining function $\rho$ with $d\rho/\rho^2$ timelike near the past cap, and, by Wick rotation, for translation-invariant metrics on $\mathbb{R}^n$ and their perturbations in the scattering category.
Problem. For each real $\lambda \ne 0$ for which these assumptions hold, must every $u \in \dot{C}^\infty(M)$ with $(\square_g - \lambda)u = 0$ vanish identically?
Source: Vasy (2020), Remark 6 of arXiv:1712.09650v1, which leaves the triviality of the nullspace of the shifted wave operator as a question. Stated by AIM — Open Applied Problems for rapidly decaying solutions, using Vasy's Proposition 7.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG, MP, SP
- Keywords
- AIM Open Applied Problems, embedded eigenvalues, Feynman propagators, Klein-Gordon operators
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{vasy2020,
author = {Vasy, Andr{\'a}s},
title = {Essential self-adjointness of the wave operator and the limiting absorption principle on {L}orentzian scattering spaces},
journal = {J. Spectr. Theory},
volume = {10},
number = {2},
year = {2020},
pages = {439--461},
doi = {10.4171/JST/301},
eprint = {1712.09650},
archivePrefix = {arXiv},
}
@misc{aim-real-energy-nullspace-on-nontrapping-lorentzian-scattering-spaces,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Real-energy nullspace on nontrapping Lorentzian scattering spaces}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/879-real-energy-nullspace-on-nontrapping-lorentzian-scattering-spaces.md},
note = {Entry 879 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01121, "Real-energy nullspace on nontrapping Lorentzian scattering spaces", clopen.solutions.