Extension of positive definite functions

Let Ω⊂Rn be an open, connected subset of Rn, and let F:Ω−Ω→C, where Ω−Ω={x−y:x,y∈Ω}, be a continuous positive definite function. We give necessary and sufficient conditions for F to have an extension to a continuous positive definite function defined on the entire Euclidean space Rn. The conditions are formulated in terms of existence of a unitary representations of Rn whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations.