On a spectral representation for correlation measures in configuration space analysis

Abstract

The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let ${\Gamma }_X$, resp.\ ${\Gamma }_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on ${\Gamma }_{X,0}$ into functions on ${\Gamma }_X$ and its adjoint $K^{\ast }$ maps probability measures on ${\Gamma }_X$ into $\sigma$-finite measures on ${\Gamma }_{X,0}$. For a probability measure $\mu$ on ${\Gamma }_X$, ${\rho }_{\mu }:=K^{\ast }\mu$ is called the correlation measure of $\mu$. We consider the inverse problem of existence of a probability measure $\mu$ whose correlation measure ${\rho }_{\mu }$ is equal to a given measure $\rho$. We introduce an operation of $\star$-convolution of two functions on ${\Gamma }_{X,0}$ and suppose that the measure $\rho$ is $\star$-positive definite, which enables us to introduce the Hilbert space ${\mathcal{H}}_{\rho }$ of functions on ${\Gamma }_{X,0}$ with the scalar product $(G^{(1)},G^{(2)}{)}_{{\mathcal{H}}_{\rho }}={\int }_{{\Gamma }_{X,0}}(G^{(1)}\star \bar{G}{}^{(2)})(\eta )\rho (d\eta )$. Under a condition on the growth of the measure $\rho$ on the $n$-point configuration spaces, we construct the Fourier transform in generalized joint eigenvectors of some special family $A=(A_ϕ)_{ϕ\in\D}$, $\D:=C_0^\infty(X)$, of commuting selfadjoint operators in ${\mathcal{H}}_{\rho }$. We show that this Fourier transform is a unitary between ${\mathcal{H}}_{\rho }$ and the $L^2$-space $L^2({\Gamma }_X,d\mu )$, where $\mu$ is the spectral measure of $A$. Moreover, this unitary coincides with the $K$-transform, while the measure $\rho$ is the correlation measure of $\mu$.