The complex-time Segal-Bargmann transform
The complex-time Segal-Bargmann transform
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复杂时间 Segal-Bargmann 变换
DOI：
10.1016/j.jfa.2019.108303
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发表时间：
2020
影响因子：
1.7
通讯作者：
Kemp, Todd
中科院分区：
文献类型：
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作者：
Driver, Bruce K.;Hall, Brian C.;Kemp, Todd
我们给出了紧型连通李群K的一种新的西格尔-巴格曼变换形式。证明了热核(ρt(X))t>；0，x∈K具有对全纯函数(ρC(τ，z))reτ>；0，z∈K C的时空解析延拓，其中K C是K的复化.新的变换定义为：积分(Bτf)(Z)=∫KρC(τ，z k−1)f(K)dk，z∈K C.0和τ∈D(S，S)(以S为中心的半径S的圆盘)，这个积分定义了KC上的一个全纯函数，每个f∈L 2(K，ρS)。我们在KC上构造了一个热核密度μS，τ使得对于所有的S，τ如上所述，B S，τ：=Bτ|L 2(K，ρS)是从L 2(K，ρS)到L 2(K，μS，τ)的全纯函数空间上的等距同构.当τ=t=S时，变换Bt，t与第二作者为紧群引入的变换一致，并由第一作者推广到紧型群.当τ=t∈(0，2 S)时，变换B S，t与前两位作者所介绍的一致。
We introduce a new form of the Segal–Bargmann transform for a connected Lie group K of compact type. We show that the heat kernel (ρ t (x)) t> 0, x∈ K has a space-time analytic continuation to a holomorphic function (ρ C (τ, z)) Re τ> 0, z∈ K C, where K C is the complexification of K. The new transform is defined by the integral (B τ f)(z)=∫ K ρ C (τ, z k− 1) f (k) d k, z∈ K C. If s> 0 and τ∈ D (s, s)(the disk of radius s centered at s), this integral defines a holomorphic function on K C for each f∈ L 2 (K, ρ s). We construct a heat kernel density μ s, τ on K C such that, for all s, τ as above, B s, τ:= B τ| L 2 (K, ρ s) is an isometric isomorphism from L 2 (K, ρ s) onto the space of holomorphic functions in L 2 (K C, μ s, τ). When τ= t= s, the transform B t, t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ= t∈(0, 2 s), the transform B s, t coincides with the one introduced by the first two authors.