Determinants of Laplacians on surfaces of finite volume
Determinants of Laplacians on surfaces of finite volume
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有限体积表面上拉普拉斯算子的行列式
DOI：
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发表时间：
1988
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通讯作者：
I. Efrat
中科院分区：
文献类型：
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作者：
I. Efrat
摘要紧流形上拉普拉斯和其他椭圆算子的行列式多年来一直是研究对象（参见 [MP、RS、Vor]）。然而，到目前为止，行列式理论尚未扩展到非紧情况，因为这些情况通常涉及离散谱和连续谱的混合。该理论的最新进展部分受到数学物理发展的推动，在紧致黎曼曲面情况下，旋量上的拉普拉斯行列式与底层曲面的 Selberg zeta 函数之间建立了联系（参见 [DP, Kie, Sar, Vor]）。我们在本文中的目的是引入非紧（有限体积）黎曼曲面上行列式的概念。这些将与按参数 (1−s) 平移的拉普拉斯 Δ 相关联，并将根据狄利克雷级数 z(w, s) 进行定义，该级数是代表离散谱和连续谱的总和。可以看出，atw=0 是有规律的，我们的主要定理（参见第 1 节）将表示 exp
$$left( {左。{ - frac{偏 }{{偏 w}}zeta (w,s)}
右|_{w = 0} }
右）$$
作为表面的 Selberg zeta 函数乘以适当的 Γ 因子。
AbstractDeterminants of the Laplace and other elliptic operators on compact manifolds have been an object of study for many years (see [MP, RS, Vor]). Up until now, however, the theory of determinants has not been extended to non-compact situations, since these typically involve a mixture of discrete and continuous spectra. Recent advances in this theory, which are partially motivated by developments in mathematical physics, have led to a connection, in the compact Riemann surface case, between determinants of Laplacians on spinors and the Selberg zeta function of the underlying surface (see [DP, Kie, Sar, Vor]).Our purpose in this paper is to introduce a notion of determinants on non-compact (finite volume) Riemann surfaces. These will be associated to the Laplacian Δ shifted by a parameters(1−s), and will be defined in terms of a Dirichlet series ζ(w, s) which is a sum that represents the discrete as well as the continuous spectrum. It will be seen to be regular atw=0, and our main theorem (see Sect. 1) will express exp
$$left( {left. { - frac{partial }{{partial w}}zeta (w,s)}
ight|_{w = 0} }
ight)$$
as the Selberg zeta function of the surface times the appropriate Γ-factor.