Resonance expansions and local-energy decay estimates for Dirac operators
Abstract
We study resonance expansions and localised long-time dynamics for three-dimensional semiclassical Dirac operators. For smooth Hermitian matrix-valued perturbations which are analytic outside a compact set, we first obtain exponential cut-off resolvent bounds away from the resonance set and polynomial bounds in thin resonance-free rectangles adjacent to the real axis. Combining these estimates with a local Oc(h−3) upper bound on the number of resonances, a pigeonhole selection of admissible contours, almost-analytic functional calculus, and a Cauchy--Green deformation, we derive resonance expansions for spectrally localised propagators. Positive-energy resonances contribute for large positive times, whereas negative-energy resonances contribute for large negative times. We then specialise to mass-type perturbations mbfβV(x) in a nontrapping energy window. Using the logarithmic resonance-free region established in our earlier work together with a quantitative FBI-transform/escape-function argument, we obtain a polynomial bound for the continued cut-off resolvent and deduce rapid local-energy decay. We subsequently refine this polynomial estimate by a finite-time propagation argument in the undistorted region. Introducing the maximal positive-sheet connection time Tχ associated with the spatial cutoff, we prove a geometric continued-resolvent estimate and, for every fixed M below the resonance-free depth constant, derive the explicit local-energy decay threshold TN>Tχ+(N+1)/M. The same analysis gives a rigorous microlocal obstruction to a uniform [hlog(1/h)]−1 bound for the full distorted resolvent and identifies a finite-flight-time amplification mechanism for the sandwiched continued resolvent at logarithmic depth.
Related
- Local Smoothing Estimates of Schrödinger Operators on Waveguides Using Refined Decoupling
- Local Energy Control in the Presence of a Zero-Energy Resonance
- Estimates of Lucas series and expansions in non-integer bases
- On asymptotic expansions of the density of states for Poisson distributed random Schrödinger operators
- Resolvent Expansions at All Spectral Thresholds for Three-Dimensional Discrete Schrödinger Operators
BibTeX
@misc{chen2026resonance,
title = {{Resonance expansions and local-energy decay estimates for Dirac operators}},
author = {Zhuo Chen and Michael Melgaard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02108},
eprint = {2610.02108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02108},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Resonance expansions and local-energy decay estimates for Dirac operators
AU - Zhuo Chen
AU - Michael Melgaard
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02108
UR - https://arxiv.org/abs/2610.02108
AB - We study resonance expansions and localised long-time dynamics for three-dimensional semiclassical Dirac operators. For smooth Hermitian matrix-valued perturbations which are analytic outside a compact set, we first obtain exponential cut-off resolvent bounds away from the resonance set and polynomial bounds in thin resonance-free rectangles adjacent to the real axis. Combining these estimates with a local Oc(h^−3) upper bound on the number of resonances, a pigeonhole selection of admissible contours, almost-analytic functional calculus, and a Cauchy--Green deformation, we derive resonance expansions for spectrally localised propagators. Positive-energy resonances contribute for large positive times, whereas negative-energy resonances contribute for large negative times. We then specialise to mass-type perturbations mbfβV(x) in a nontrapping energy window. Using the logarithmic resonance-free region established in our earlier work together with a quantitative FBI-transform/escape-function argument, we obtain a polynomial bound for the continued cut-off resolvent and deduce rapid local-energy decay. We subsequently refine this polynomial estimate by a finite-time propagation argument in the undistorted region. Introducing the maximal positive-sheet connection time T_χ associated with the spatial cutoff, we prove a geometric continued-resolvent estimate and, for every fixed M below the resonance-free depth constant, derive the explicit local-energy decay threshold T_N>T_χ+(N+1)/M. The same analysis gives a rigorous microlocal obstruction to a uniform [hlog(1/h)]^−1 bound for the full distorted resolvent and identifies a finite-flight-time amplification mechanism for the sandwiched continued resolvent at logarithmic depth.
ER -