Eigenfunctions of annihilation operators associated with Wigner's commutation relations

Abstract

We consider linear continuous operators acting on the space of all entire functions with the uniform convergence topology and satisfying Wigner's commutation relations. These operators are closely connected with the Dunkl generalized convolution operators. We study the problem of description of eigenfunctions of these operators. It is shown that under some conditions the eigenfunctions of the operator under study can be described by Dunkl generalized translates of entire functions belonging to it's kernel. We also discuss the problem of completeness of the systems of eigenfunctions.