Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case

Abstract

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 $\mathcal{L}={\partial }_x^2+\frac{1}{2x}{\partial }_x+\frac{1}{x}{\partial }_{\theta }^2$ 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.

33 pages

References in corpus (1)

• Sturm-Liouville hypergroups without the compactness axiom