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 $\Dl_j=\Dl_0+\mbfβv_j$, $j=1,2$, satisfying the exterior analyticity, decay and non-degeneracy assumptions of the paper, we associate a relative distribution $\mu$ 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 $C{\hslash }^{-3}$ resonances of the two operators, counted with multiplicity. The proof transfers an analytic wave-front singularity of $\mu$ 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 $C{\hslash }^{-3}$ resonances in every complex neighbourhood of $E_0=1+\sup v$ and of $-E_0$, for all sufficiently small $\hslash$. 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.