Abstract. In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.
Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel function of this integral transform is constructed using the Mehler formula. An explicit expression for the integral transform is obtained and allows for the study of its properties. In the second part, two kinds of corresponding convolutions are examined: Mustard convolutions and convolutions based on generalised translation operators. The paper finishes by demonstrating the connection between both.
The Segal-Bargmann transform is a unitary map between the Schrödinger and the Fock space, which is e.g. used to show the integrability of quantum Rabi models. Slice monogenic functions provide the framework in which functional calculus for quaternionic quantum mechanics can be developed. In this paper, a generalisation of the Segal-Bargmann transform to the context of slice monogenic functions is constructed and studied in detail. It is shown to interact appropriately with the recently constructed slice Fourier transform. This leads furthermore to a construction of a slice Fock space, which is shown to be a reproducing kernel space.
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