论文标题
半古典质量渐近学有关固定空间的
Semi-classical mass asymptotics on stationary spacetimes
论文作者
论文摘要
我们研究频谱$ \ {λ_j（m）\} _ {j = 1}^{\ infty} $ simelike Killing杀伤矢量field $ z $，用作较大的klein-gordon方程$（\ boge_g + m m^2）a y Hilbert Space of Hilbert shielt of Hilbert Space in Hilbert Space上的差异操作员$ d_Z $(M, g)$ with compact Cauchy hypersurface.反质量$ m^{ - 1} $正式就像schrödinger方程中的普朗克常数一样，我们将weyl渐近学作为$ m \ to \ infty $，用于$ n_ {ν，c}，c}，c}（m）= \＃\ \ \ { \frac{C}{m}, ν+ \frac{C}{m} ]\}$$ for a given $C > 0$.半古典质量渐近学受杀伤流的动力学$ e^{tz} $的动力，在质量$ 1 $ 1 $ GEODESICS $γ$的空间中，其中$ \ langle \dotγ，z \ \ w \ rangle =ν$。
We study the spectrum $\{λ_j(m)\}_{j=1}^{\infty}$ of a timelike Killing vector field $Z$ acting as a differential operator $D_Z$ on the Hilbert space of solutions of the massive Klein-Gordon equation $(\Box_g + m^2) u = 0$ on a globally hyperbolic stationary spacetime $(M, g)$ with compact Cauchy hypersurface. The inverse mass $m^{-1}$ is formally like the Planck constant in a Schrödinger equation, and we give Weyl asymptotics as $m \to \infty$ for the number $$N_{ν, C}(m)= \# \{j \mid \frac{λ_j(m)}{m} \in [ν- \frac{C}{m}, ν+ \frac{C}{m} ]\}$$ for a given $C > 0$. The semi-classical mass asymptotics are governed by the dynamics of the Killing flow $e^{tZ} $ on the hypersurface in the space of mass $1$ geodesics $γ$ where $\langle \dotγ, Z \rangle= ν$.
