论文标题
无质量狄拉克操作员和光谱渐近学的全球传播器
Global propagator for the massless Dirac operator and spectral asymptotics
论文作者
论文摘要
我们在封闭的Riemannian 3 manifold上构建了无质量狄拉克运算符$ w $的传播器，作为两个不变定义的振荡积分的总和，具有杰出的复杂价值相位函数。两个振荡积分 - 正和负繁殖器 - 分别对应于$ W $的正和负特征值。这使我们能够为传播器的完整符号（Cotangent Bundle上的标量矩阵函数）提供全局不变的定义，主要符号的封闭公式以及用于显式计算其所有同质组件的算法。此外，我们从几何不变剂方面获得了繁殖者的主要和次主符号的少量扩展。最后，我们使用结果来计算$ w $的特征值计数函数的渐近扩展中的第三个局部Weyl系数。
We construct the propagator of the massless Dirac operator $W$ on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals -- the positive and the negative propagators -- correspond to positive and negative eigenvalues of $W$, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of $W$.
