On rank one Riemannian symmetric spaces of noncompact type, which accommodates all hyperbolic spaces, we show that characterization of the eigenfunctions/eigendistributions of the Laplace–Beltrami operator, through the action of the heat operator is possible only when we confine in a small time. The results are different from their counter parts in the Euclidean spaces. All results and their proofs extend to Damek–Ricci spaces.

中文翻译：

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拉普拉斯-贝尔特拉米算子的特征函数通过短时间的热传播表征

在容纳所有双曲空间的非紧型的一阶黎曼对称空间上，我们证明了 Laplace-Beltrami 算子的特征函数/特征分布的表征，通过热算子的作用，只有当我们限制在很短的时间内是可能的。结果与其在欧几里得空间中的对应部分不同。所有结果及其证明都扩展到 Damek-Ricci 空间。