论文标题
旋转狄拉克操作员的Selberg Trace公式在退化双曲线表面
The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces
论文作者
论文摘要
我们研究了双曲线表面家族的自旋狄拉克操作员的频谱，其中一组不相交的简单的大地测量学缩小至$ 0 $，这是沿着每个捏缝的大地测量的旋转结构是不平凡的。主要工具是在有限区域双曲线表面上的狄拉克操作员的痕量公式。我们得出了Huber定理的一种版本，并为双尖的双曲线表面提供了非标准的小热量渐近膨胀。作为推论，我们为狄拉克操作员的特征值找到了同时的Weyl定律，该定律在退化参数中是均匀的。主要的结果是与此类双曲表面家族有关的Selberg Zeta函数的收敛性。 $ \ {\ pm 1 \} $ - 有价值的类函数$ \ varepsilon $由旋转结构确定。
We investigate the spectrum of the spin Dirac operator on families of hyperbolic surfaces where a set of disjoint simple geodesics shrink to $0$, under the hypothesis that the spin structure is non-trivial along each pinched geodesic. The main tool is a trace formula for the Dirac operator on finite area hyperbolic surfaces. We derive a version of Huber's theorem and a non-standard small-time heat trace asymptotic expansion for hyperbolic surfaces with cusps. As a corollary we find a simultaneous Weyl law for the eigenvalues of the Dirac operator which is uniform in the degenerating parameter. The main result is the convergence of the Selberg zeta function associated to the Dirac operator on such families of hyperbolic surfaces. A central role is played by a $\{ \pm 1 \}$-valued class function $\varepsilon$ determined by the spin structure.
