Quantum analogues of white noise delta functions

Abstract

In this paper, we construct a quantum analogue of a white noise delta function. For our main purpose, we first discuss several white noise delta functions, specially, an infinite-dimensional analogue of Donsker’s delta function. Then as a general setting, we construct white noise delta functions on white noise test functionals and we derive a Cauchy problem whose solution is given by the white noise delta function. Also, we prove that the white noise delta function is an eigenvector of the exotic Laplacian. By applying the canonical topological isomorphisms between the spaces of white noise operators and the spaces of white noise functionals, we formulate a quantum analogue of the white noise delta function which satisfies a quantum white noise differential equation associated with the quantum Volterra Laplacian. Finally, we prove that quantum analogue of the white noise delta function is an eigenoperator of the quantum exotic Laplacian.