论文标题
二维拉普拉斯 - 贝特拉米操作员的产品公式和卷积：超越琐事
Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case
论文作者
论文摘要
我们介绍了与给定椭圆偏差操作员相关的卷积操作员家族的概念。这种卷积结构被证明是一类带有锥形指标的二维歧管上的一类拉普拉斯 - 贝特拉米操作员存在的卷积结构。这种结构引起了Laplace-Beltrami操作员产生的Markovian Semigroup的卷积半群代表。 在操作员的特殊情况下并且反转公式可以以汇合的超几何函数为封闭形式。本文的结果可以解释为对多参数特征值问题框架的一维广义卷积理论的自然扩展。
We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional manifolds endowed with cone-like metrics. This structure gives rise to a convolution semigroup representation for the Markovian semigroup generated by the Laplace-Beltrami operator. In the particular case of the operator $\mathcal{L} = \partial_x^2 + {1 \over 2x} \partial_x + {1 \over x} \partial_θ^2$ on $\mathbb{R}^+ \times \mathbb{T}$, we deduce the existence of a convolution structure for a two-dimensional integral transform whose kernel and inversion formula can be written in closed form in terms of confluent hypergeometric functions. The results of this paper can be interpreted as a natural extension of the theory of one-dimensional generalized convolutions to the framework of multiparameter eigenvalue problems.
