关于库仑散射与双曲散射之间 intertwining 映射的探讨
On the intertwining map between Coulomb and hyperbolic scattering
AI总结：
本文提出一个将库仑散射与双曲散射间本征函数空间联系起来的单位元算子，推广了福克变换到连续谱，并解释了两种设置中散射矩阵相同的原因。
摘要：
我们构建了一个在库仑算子和双曲空间拉普拉斯-贝尔特里算子的广义本征函数的希尔伯特空间之间的作用算子，该算子在 $L^2(\mathbb{S}^{d-1})$ 上 intertwines 它们各自的泊松算子。所构建的算子将福克的单位元变换推广到连续谱的情形，该变换最初定义在吸引性库仑算子的离散谱和球面上的拉普拉斯-贝尔特里算子的离散谱之间。此外，该映射解释了在两种不同设置中散射矩阵为何相同，并提供了库仑哈密顿量泊松算子的显式公式。
英文摘要：
We construct a unitary operator between Hilbert spaces of generalized eigenfunctions of Coulomb operators and the Laplace-Beltrami operator of hyperbolic space that intertwines their respective Poisson operators on $L^2(\mathbb{S}^{d-1})$. The constructed operator generalizes Fock's unitary transformation, originally defined between the discrete spectra of the attractive Coulomb operator and the Laplace-Beltrami operator on the sphere, to the setting of continuous spectra. Among other connections, this map explains why the scattering matrices are the same in these two different settings, and it also provides an explicit formula for the Poisson operator of the Coulomb Hamiltonian.
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