Narrow Results

Every Evans-Hudson flow on the algebra of all bounded operators on a Hilbert space leads to a semigroup of *-endomorphisms, and then to a continuous tensor product system of Hilbert spaces. Here we have a new representation for exponential (Boson Fock space) product systems. This helps us to show that only such product systems arise from one-dimensional Evans-Hudson flows.

The main purpose of this paper is to derive a general structure of Gegenbauer white noise analysis as a counterpart class of non-Lévy white noise. Namely, we consider, on an appropriate space of distributions, , a Gegenbauer white noise measure, , and construct a nuclear triple of test and generalized functions. A basic role is played by the chaos expansion. By using the S_β-transform we prove a general characterization theorems for Gegenbauer white noise distributions, white noise test functions in terms of analytical functions.

We describe the nontrivial central extensions CE(Heis) of the Heisenberg algebra and their representation as sub–algebras of the Scheoedinger algebra. We also present the characteristic and moment generating functions of the random variable corresponding to the self-adjoint sum of the generators of CE(Heis).

We formulate a class of differential equations for white noise operators including the quantum white noise derivatives and obtain a general form of the solutions. As application we characterize intertwining operators appearing in the implementation problem for the canonical commutation relation.