Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I
Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I
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非欧空间中波动方程自守解的平移表示 I。
DOI：
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
R. Phillips
中科院分区：
文献类型：
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作者：
P. Lax;R. Phillips
本文研究了n维双曲空间H_n中Laplace-Beltrami算子Δ作用于自守函数的谱理论。研究了具有有限边数和无限体积的基本多面体F的离散子群Γ。关于这些，我们以前已经证明了Δ的谱至多包含[-（1/2（n - 1））2，0]中有限个点本征值，并且不少于（1/2（n-1））2。本文证明了Δ的谱在（-∞，-（1/2（n - 1））2）中是绝对连续的，且具有无穷重数.我们的方法使用Faddeev和Pavlov引入的非欧几里德波动方程，
能量EF被定义为（ut，ut）-（u，Lu），其中括号是基本多面体上关于双曲度量的不变体积的L2标积。在将初始数据与时间t处的数据相关联的算子U（t）的组下能量守恒。我们用L2（R，N）构造了自守数据空间的两个等距表示，它将U（t）的作用转化为平移。这些表示明确给出的积分的数据horospheres。在第二部分中，我们将展示这些表示的完整性。
utt-Lu = 0，L = Δ +（1/2（n - 1））2.
This paper deals with the spectral theory of the Laplace-Beltrami operator Δ acting on automorphic functions in n-dimensional hyperbolic space Hn. We study discrete subgroups Γ which have a fundamental polyhedron F with a finite number of sides and infinite volume. Concerning these we have shown previously that the spectrum of Δ contains at most a finite number of point eigenvalues in [-(1/2(n - 1))2, 0], and none less than (1/2(n -1))2. Here we prove that the spectrum of Δ is absolutely continuous and of infinite multiplicity in (-∞, -(1/2(n - 1))2). Our approach uses the non-Euclidean wave equation introduced by Faddeev and Pavlov,
Energy EF is defined as (ut, ut)-(u, Lu), where the bracket is the L2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric. Energy is conserved under the group of operator U(t) relating initial data to data at time t. We construct two isometric representations of the space of automorphic data by L2(R, N) which transmute the action of U(t) into translation. These representations are given explicitly in terms of integrals of the data over horospheres. In Part II we shall show the completeness of these representations.
utt-Lu = 0, L = Δ + (1/2(n - 1))2.