The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds
Abstract
A central problem in harmonic analysis is to understand how frequency resolution constrains stability under nonlinear changes of coordinates. Motivated by Mallat's wavelet scattering theory, we establish a sharp resolution--robustness principle for nonlinear representations built from two-dimensional wave-packet systems, showing that the transverse geometry of the frequency channels determines the critical Sobolev regularity. Whenever the channels are thinner than in the wavelet regime, arbitrarily small smooth deformations can move high-frequency mass between adjacent channels, producing instability already at the level of the linear filter bank and the first scattering layer. We match this obstruction with scale-sharp commutator estimates propagated through the full scattering cascade, obtaining Lipschitz stability at the critical regularity threshold. Together, the results provide the first intrinsic deformation-stability theory for non-wavelet Euclidean scattering transforms and identify a microlocal transport mechanism governing robustness across nonlinear multiscale representations.
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BibTeX
@misc{getter2026stability,
title = {{The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds}},
author = {Max Getter and S. Ivan Trapasso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.21578},
eprint = {2607.21578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21578},
}
RIS
TY - UNPB
TI - The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds
AU - Max Getter
AU - S. Ivan Trapasso
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2607.21578
UR - https://arxiv.org/abs/2607.21578
AB - A central problem in harmonic analysis is to understand how frequency resolution constrains stability under nonlinear changes of coordinates. Motivated by Mallat's wavelet scattering theory, we establish a sharp resolution--robustness principle for nonlinear representations built from two-dimensional wave-packet systems, showing that the transverse geometry of the frequency channels determines the critical Sobolev regularity. Whenever the channels are thinner than in the wavelet regime, arbitrarily small smooth deformations can move high-frequency mass between adjacent channels, producing instability already at the level of the linear filter bank and the first scattering layer. We match this obstruction with scale-sharp commutator estimates propagated through the full scattering cascade, obtaining Lipschitz stability at the critical regularity threshold. Together, the results provide the first intrinsic deformation-stability theory for non-wavelet Euclidean scattering transforms and identify a microlocal transport mechanism governing robustness across nonlinear multiscale representations.
ER -