[Paper Review] On a spectral representation for correlation measures in configuration space analysis

Yu. M. Berezansky, Yu. G. Kondratiev+2 more|ArXiv.org|Aug 14, 2006

Morphological variations and asymmetry14 references21 citations

TL;DR

This paper establishes a spectral representation for correlation measures in configuration space analysis by introducing a $∞star$-convolution structure on finite configurations, enabling the construction of a Hilbert space ${\cal H}_\rho$ where a family of commuting selfadjoint operators admits a Fourier transform via the projective spectral theorem. The key result is that this transform yields a unitary map from ${\cal H}_\rho$ to $L^2(\Gamma_X, \mu)$, with the measure $\mu$ being the spectral measure and $\rho$ its correlation measure, thus solving the inverse problem of reconstructing a probability measure from its correlation measure under growth conditions.

ABSTRACT

The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let $Γ_X$, resp.\ $Γ_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on $Γ_{X,0}$ into functions on $Γ_{X}$ and its adjoint $K^*$ maps probability measures on $Γ_X$ into $σ$-finite measures on $Γ_{X,0}$. For a probability measure $μ$ on $Γ_X$, $ρ_μ:=K^*μ$ is called the correlation measure of $μ$. We consider the inverse problem of existence of a probability measure $μ$ whose correlation measure $ρ_μ$ is equal to a given measure $ρ$. We introduce an operation of $\star$-convolution of two functions on $Γ_{X,0}$ and suppose that the measure $ρ$ is $\star$-positive definite, which enables us to introduce the Hilbert space ${\cal H}_ρ$ of functions on $Γ_{X,0}$ with the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_ρ}= \int_{Γ_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(η) ρ(dη)$. Under a condition on the growth of the measure $ρ$ 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 ${\cal H}_ρ$. We show that this Fourier transform is a unitary between ${\cal H}_ρ$ and the $L^2$-space $L^2(Γ_X,dμ)$, where $μ$ is the spectral measure of $A$. Moreover, this unitary coincides with the $K$-transform, while the measure $ρ$ is the correlation measure of $μ$.

Motivation & Objective

• To solve the inverse problem of reconstructing a probability measure $\mu$ on the configuration space $\Gamma_X$ from a given $\sigma$-finite measure $\rho$ on $\Gamma_{X,0}$.
• To establish a spectral-theoretic framework for correlation measures using the $K$-transform and its adjoint $K^*$.
• To show that under a growth condition on $\rho$, the $\star$-convolution structure on $\Gamma_{X,0}$ allows for a unitary Fourier transform to $L^2(\Gamma_X, \mu)$.
• To prove that the $K$-transform coincides with the Fourier transform in generalized joint eigenvectors of a commuting family of operators in ${\cal H}_\rho$, thereby identifying $\mu$ as the spectral measure.

Proposed method

• Introduce a $\star$-convolution operation on functions over $\Gamma_{X,0}$, defined via the $K$-transform and its adjoint.
• Define a Hilbert space ${\cal H}_\rho$ with inner product $(G^{(1)}, G^{(2)})_{{\cal H}_\rho} = \int_{\Gamma_{X,0}} (G^{(1)} \star \overline{G}^{(2)})(\eta) \, \rho(d\eta)$, using the $\star$-positive definiteness of $\rho$.
• Construct a family of operators $(A_\varphi)_{\varphi \in {\cal D}}$ on ${\cal H}_\rho$ via $A_\varphi G(\eta) = (\varphi \star G)(\eta)$, where $\varphi \in C_0^\infty(X)$.
• Prove that the operators $A_\varphi$ are essentially selfadjoint and their closures form a commuting family of selfadjoint operators.
• Apply the projective spectral theorem to the commuting family $(A_\varphi^\sim)$, yielding a unitary Fourier transform from ${\cal H}_\rho$ to $L^2({\cal D}^\prime, \mu)$.
• Show that this Fourier transform coincides with the $K$-transform, and that $\mu$ is the spectral measure with $\rho$ as its correlation measure.

Experimental results

Research questions

• RQ1 Under what conditions on a $\sigma$-finite measure $\rho$ on $\Gamma_{X,0}$ does there exist a probability measure $\mu$ on $\Gamma_X$ such that $\rho = K^*\mu$?
• RQ2 Can the $K$-transform be interpreted as a Fourier transform in generalized joint eigenvectors of a commuting family of operators in a Hilbert space constructed from $\rho$?
• RQ3 Does the $\star$-convolution structure on $\Gamma_{X,0}$ allow for a spectral representation of $\rho$ via the projective spectral theorem?
• RQ4 Is the resulting measure $\mu$ on $\Gamma_X$ supported on the full configuration space $\Gamma_X$ under the growth condition on $\rho$?
• RQ5 Can the $K$-transform be realized as a unitary operator between ${\cal H}_\rho$ and $L^2(\Gamma_X, \mu)$, with $\mu$ being the spectral measure of the operator family?

Key findings

• The $K$-transform is unitarily equivalent to the Fourier transform in generalized joint eigenvectors of the commuting selfadjoint operators $(A_\varphi^\sim)$, establishing a spectral representation of the $K$-transform.
• The Hilbert space ${\cal H}_\rho$ is constructed from the $\star$-convolution and $\rho$-positive definiteness, ensuring a well-defined inner product.
• The spectral measure $\mu$ of the family $(A_\varphi^\sim)$ satisfies $\mu(\Gamma_X) = 1$, meaning $\mu$ is supported on the full configuration space.
• The measure $\rho$ is the correlation measure of $\mu$, i.e., $\rho = K^*\mu$, confirming the inverse problem is solved.
• Under the growth condition $\rho(\Gamma_\Lambda^{(n)}) \leq (2 + \varepsilon)^n$, the measure $\widetilde{\mu}_\Lambda$ is a probability measure on $\Gamma_\Lambda$, and $\mu$ is consistent across compact subsets.
• The Fourier transform of $\mu_\Lambda$ extends analytically to a neighborhood of zero, and the characteristic function of $\mu_\Lambda$ matches the generating function of $\rho$ via the $K$-transform.