论文标题
复杂双曲空间中晶格点的积分公式和渐近行为
Integral Formulas and asymptotic behavior of lattice points in complex hyperbolic space
论文作者
论文摘要
本文处理了复杂的双曲线空间$ \ mathbb {c} h^n $中的$γ$ lattice点问题$γ$相关的$γ$ lattice点问题。我们为$ n（t，z，z，z'）$ $γ$的局部平均值提供了两个积分公式 - 在$ \ mathbb {c} h^n $的半径$ t $ in radius $ t $中的晶格点。首先是根据$ \ Mathbb {C} h^n $在$γ$ -Automormorphic Wave方程的解决方案方面，第二个是根据Laplace-Beltrami操作员在$γ$ -Automorphic边界条件下的光谱功能给出的。我们使用获得的积分公式来获得数字$ n（t，z，z'）$作为$ t \ rightarrow \ infty $的渐近行为，并估计其余期限。我们的主要工具是在$γ$自动形界边界条件下的Laplace-Beltrami操作员的复杂双曲空间，特殊功能和光谱理论上的波动方程的明确解。
This paper deals with the $Γ$-lattice points problem associated to a discrete subgroup of motions $Γ$ in the complex hyperbolic space $\mathbb{C} H^n$. We give two integral formulas for the local average of the number $N(T, z, z')$ of $Γ$- lattice points in a sphere of radius $T$ in $\mathbb{C} H^n$. The first on is in terms of the solution of the $Γ$-automorphic wave equation on $\mathbb{C} H^n$ and the second is given in terms of the spectral function of the Laplace-Beltrami operator under $Γ$-automorphic boundary conditions. We use the obtained integral formulas to obtain an asymptotic behavior of the number $N(T, z, z')$ as $T\rightarrow \infty$, with an estimate of the remainder term. Our principal tools are the explicit solution of the wave equation on the complex hyperbolic space, special functions and spectral theory of the Laplace-Beltrami operator under $Γ$ automorphic boundary conditions.
