On a spectral representation for correlation measures in configuration space analysis

TL;DR

This paper develops a spectral representation for correlation measures in configuration space analysis, linking the $K$-transform with a Fourier transform in a Hilbert space framework, enabling the inverse problem of measure reconstruction.

Contribution

It introduces a spectral approach using the projective spectral theorem to analyze correlation measures and connects the $K$-transform with a Fourier transform in a Hilbert space setting.

Findings

Constructed a Fourier transform as a unitary operator

Established the $K$-transform as a Fourier transform

Connected correlation measures with spectral measures of operators

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 $K^*$ maps probability measures on $\Gamma_X$ into $\sigma$-finite measures on $\Gamma_{X,0}$. For a probability measure $\mu$ on $\Gamma_X$, $\rho_\mu:=K^*\mu$ 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 ${\cal H}_\rho$ of functions on $\Gamma_{X,0}$ with the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_{\rho}}= \int_{\Gamma_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(\eta) \rho(d\eta)$. 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 $A=(A_\phi)_{\phi\in\D}$, $\D:=C_0^\infty(X)$, of commuting selfadjoint operators in ${\cal H}_\rho$. We show that this Fourier transform is a unitary between ${\cal H}_{\rho}$ 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 $\mu$.