Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a spectral representation for correlation measures in configuration
  space analysis   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 $\mu$.

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The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
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The categories of this paper are: cs
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn