Harmonic analysis on hyperboloids
Harmonic analysis on hyperboloids
复制标题
双曲面的调和分析
DOI：
10.1016/0022-1236(73)90001-3
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发表时间：
1973
影响因子：
1.7
通讯作者：
R. Strichartz
中科院分区：
文献类型：
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作者：
R. Strichartz
将作用于l2的O （n, n）的正则表示(O （n, n) O (n, n−1)）分解为不可约表示的直接积分。齐次空间O (n, n) O （n, n−1）被实现为双曲面H={(x, t) λ R n+ n: γ γ γ 2−γ γ γ 2= 1}。这个问题本质上等价于求出h上某个自伴随不变微分算子□h的谱分辨率，它是R n+ n上算子□= Δ x−Δ t的切向部分。□h的谱包含一个离散部分（除n = 1时外），其特征函数是由□u= 0的h解产生的，该解在t < x >区域内消失。作为O （n, n）的表示，H−⊕H−酉等价于锥{(x, t): γ γ γ = γ γ γ}在l2上的正则表示，并通过在给定锥上的边值下解方程□u= 0得到交织算子。给出了谱分解的显式公式。特殊情况n= n= 2给出了SL （2， R）的Plancherel公式。
The regular representation of O (n, N) acting on L 2 (O (n, N) O (n, N− 1)) is decomposed into a direct integral of irreducible representations. The homogeneous space O (n, N) O (n, N− 1) is realized as the Hyperboloid H={(x, t) ϵ R n+ N:¦ t¦ 2−¦ x¦ 2= 1}. The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator□ h on H, which is the tangential part of the operator□= Δ x− Δ t on R n+ N. The spectrum of□ h contains a discrete part (except when N= 1) with eigenfunctions generated by restricting to H solutions of□ u= 0 which vanish in the region¦ t¦<¦ x¦, and a continuous part H−. As a representation of O (n, N), H−⊕ H− is unitarily equivalent to the regular representation on L 2 of the cone {(x, t):¦ x¦ 2=¦ t¦ 2}, and the intertwining operator is obtained by solving the equation□ u= 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n= N= 2 gives the Plancherel formula for SL (2, R).