Johnsen, J.
ON THE SPECTRAL PROPERTIES OF WITTEN-LAPLACIANS, THEIR RANGE PROJECTIONS AND
BRASCAMP--LIEB'S INEQUALITY
(508K, Post Script)
ABSTRACT. A study is made of a recent integral identity of B.~Helffer and
J.~Sj{\"o}strand, which for a not yet fully determined class of probability
measures yields a formula for the covariance of two functions (of
a stochastic variable); in comparison with the
Brascamp--Lieb inequality, this formula is a more flexible and in some contexts
stronger means for the analysis of correlation asymptotics in statistical
mechanics. Using a fine version of the Closed Range Theorem, the identity's
validity is shown to be equivalent to some explicitly given
spectral properties of Witten--Laplacians on Euclidean space, and
the formula is moreover deduced from the obtained abstract expression
for the range projection. As a corollary, a generalised version of
Brascamp--Lieb's inequality is obtained. For a certain class of
measures occuring in statistical mechanics, explicit criteria for the
Witten-Laplacians are found from the Persson--Agmon formula, from
compactness of embeddings and from the
Weyl calculus, which give results for closed range, strict positivity,
essential self-adjointness and domain characterisations.