  Due to its intimate relation to Spectral Theory and Schr\"{o}dinger
operators, the multivariate moment problem has been a subject of many
researches, so far without essential success (if one compares with the
one--dimensional case). In the present paper we reconsider a basic axiom of the
standard approach - the positivity of the measure. We introduce the so--called
pseudopositive measures instead. One of our main achievements is the solution
of the moment problem in the class of the pseudopositive measures. A measure \
$\mu$ is called pseudopositive if its Laplace-Fourier coefficients
$\mu_{k,l}(r) ,$ $r\geq0,$ in the expansion in spherical harmonics are
non--negative. Another main profit of our approach is that for pseudopositive
measures we may develop efficient ''cubature formulas'' by generalizing the
classical procedure of Gauss--Jacobi: for every integer \ $p\geq1$ we construct
a new pseudopositive measure $\nu_{p}$ having ''minimal support'' and such that
$\mu(h) =\nu_{p}(h) $ for every polynomial $h$ with $\Delta^{2p}h=0.$ The proof
of this result requires application of the famous theory of Chebyshev, Markov,
Stieltjes, Krein for extremal properties of the Gauss-Jacobi measure, by
employing the classical orthogonal polynomials $p_{k,l;j},$ $j\geq0,$ with
respect to every measure $\mu_{k,l}.$ As a byproduct we obtain a notion of
multivariate orthogonality defined by the polynomials $p_{k,l;j}$. A major
motivation for our investigation has been the further development of new models
for the multivariate Schr\"{o}dinger operators, which generalize the classical
result of M. Stone saying that the one--dimensional orthogonal polynomials
represent a model for the self--adjoint operators with simple spectrum.
