尖端平坦酉线丛上的伪拉普拉斯算子：Alvarez-Wentworth边界条件
Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Alvarez--Wentworth boundary conditions
摘要：
尖端是一种度量奇点，被描述为$S^1 \times \left] a, +\infty \right[$与庞加莱度量的乘积。底层集合也可以被看作是服从于转换$T : \left( x,y \right) \longrightarrow \left( x+1, y \right)$动作的$\mathbb{R} \times \left] a, +\infty \right[$。在它上面，我们可以考虑一个全纯线丛$L$，它来自由$T$生成的群的酉特征标。复数模在$L$上导出了一个平坦度量，并且作用于函数的伪拉普拉斯$Δ_{L,0}$可以与Chern联络相关联.需要指定边界条件，这里选择的是Alvarez-Wentworth边界条件，它是Dirichlet和Neumann边界条件的组合。本文的目的是找到zeta-正则行列式$\det \left( Δ_{L,0} + μ\right)$的渐近行为，当$μ> 0$对任何$a$趋于无穷大时，以及当$a$对$μ= 0$趋于无穷大时。
英文摘要：
A cuspidal end is a type of metric singularity, described as a product $S^1 \times \left] a, +\infty \right[$ with the Poincaré metric. The underlying set can also be seen as $\mathbb{R} \times \left] a, +\infty \right[$ subject to the action of the translation $T : \left( x,y \right) \longrightarrow \left( x+1, y \right)$. On it, one may consider a holomorphic line bundle $L$, coming from a unitary character of the group generated by $T$. The complex modulus induces a flat metric on $L$, and a pseudo-Laplacian $Δ_{L,0}$ acting on functions can be associated to the Chern connection. One needs to specify boundary conditions, and they are here chosen to be the Alvarez--Wentworth boundary conditions, which are a combination of Dirichlet and Neumann boundary conditions. The aim of this paper is to find the asymptotic behavior of the zeta-regularized determinant $\det \left( Δ_{L,0} + μ\right)$, as $μ> 0$ goes to infinity for any $a$, and also as $a$ goes to infinity for $μ= 0$.
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