发表机构
University of Sussex（萨塞克斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对三维半经典Dirac算子，在质量型势下证明每个复邻域内共振数至少为Cħ⁻³，与上界匹配，并给出相对迹公式。
AI 中文摘要
我们建立了具有质量型势的三维半经典Dirac算子共振数的下界。对于满足本文外部解析性、衰减和非退化假设的一对算子$\Dl_j=\Dl_0+\mbf{\beta}v_j$（$j=1,2$），我们将一个相对分布$\mu$关联到相应的质量函数。我们证明，如果$E_0>1$是$\supp\mu$的边界点，那么$E_0$的每个复邻域以及$-E_0$的每个复邻域中，两个算子合计至少包含$C\hbar^{-3}$个共振（按重数计）。证明将$\mu$的解析波前奇异性转移到相对经典态密度，并将其与Khochman的局部迹公式和相对半经典迹公式相结合。作为推论，如果$v$是单个有界质量型势，在外部扇形中解析且具有幂衰减，满足$\inf(1+v)>0$和$\sup v>0$，则对于所有足够小的$\hbar$，$\Dl=\Dl_0+\mbf{\beta}v$在$E_0=1+\sup v$和$-E_0$的每个复邻域中至少具有$C\hbar^{-3}$个共振。该指数与Khochman的上界一致，两个共振族通过谱对称性相关联，且$E_0$必然是俘获能量。我们还证明了论证中所需的一般厄米矩阵值扰动的相对半经典迹公式。
英文摘要
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 $C\hbar^{-3}$ 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 $C\hbar^{-3}$ resonances in every complex neighbourhood of $E_0=1+\sup v$ and of $-E_0$, for all sufficiently small $\hbar$. 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.
