Lower bound on Dirac resonances with mass-type potentials in the semi-classical limit
Abstract
We establish lower bounds on the number of resonances for three-dimensional semiclassical Dirac operators with mass-type potentials. For a pair Dlj=Dl0+mbfβvj, j=1,2, satisfying the exterior analyticity, decay and non-degeneracy assumptions of the paper, we associate a relative distribution μ to the corresponding mass functions. We prove that if E0>1 is a boundary point of suppμ, then every complex neighbourhood of E0, and likewise of −E0, contains in total at least Cℏ−3 resonances of the two operators, counted with multiplicity. The proof transfers an analytic wave-front singularity of μ to the relative classical density of states and combines this with Khochman's local trace formula and a relative semiclassical trace formula. As a consequence, if v is a single bounded mass-type potential, analytic with power decay in an exterior sector, with inf(1+v)>0 and sup v>0, then Dl=Dl0+mbfβv has at least Cℏ−3 resonances in every complex neighbourhood of E0=1+sup v and of −E0, for all sufficiently small ℏ. The exponent agrees with Khochman's upper bound, the two resonance families are related by spectral symmetry, and E0 is necessarily a trapping energy. We also prove the relative semiclassical trace formula needed in the argument for general Hermitian matrix-valued perturbations.
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BibTeX
@misc{chen2026lower,
title = {{Lower bound on Dirac resonances with mass-type potentials in the semi-classical limit}},
author = {Zhuo Chen and Michael Melgaard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02356},
eprint = {2610.02356},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02356},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Lower bound on Dirac resonances with mass-type potentials in the semi-classical limit
AU - Zhuo Chen
AU - Michael Melgaard
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02356
UR - https://arxiv.org/abs/2610.02356
AB - We establish lower bounds on the number of resonances for three-dimensional semiclassical Dirac operators with mass-type potentials. For a pair Dl_j=Dl_0+mbfβv_j, j=1,2, satisfying the exterior analyticity, decay and non-degeneracy assumptions of the paper, we associate a relative distribution μ to the corresponding mass functions. We prove that if E_0>1 is a boundary point of suppμ, then every complex neighbourhood of E_0, and likewise of −E_0, contains in total at least Cℏ^−3 resonances of the two operators, counted with multiplicity. The proof transfers an analytic wave-front singularity of μ to the relative classical density of states and combines this with Khochman's local trace formula and a relative semiclassical trace formula. As a consequence, if v is a single bounded mass-type potential, analytic with power decay in an exterior sector, with inf(1+v)>0 and sup v>0, then Dl=Dl_0+mbfβv has at least Cℏ^−3 resonances in every complex neighbourhood of E_0=1+sup v and of −E_0, for all sufficiently small ℏ. The exponent agrees with Khochman's upper bound, the two resonance families are related by spectral symmetry, and E_0 is necessarily a trapping energy. We also prove the relative semiclassical trace formula needed in the argument for general Hermitian matrix-valued perturbations.
ER -