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Correlation asymptotics and Witten Laplacians. (English) Zbl 0877.35084

In an earlier work [J. Stat. Phys. 74, 349-409 (1994)], B. Helffer and the author studied correlations associated with the measure ${e}^{-2\phi \left(x\right)/h}dx$ on ${ℝ}^{m}$, where $\phi$ is a smooth convex function, with special attention to the limit as $m$ tends to infinity. We observed (under suitable assumptions) that the expectation $〈u〉$ of a function $u\left(x\right)$ can be obtained by solving the equation

$u-〈u〉=\left(-{h}^{2}{\Delta }+2\nabla \phi ·h{\partial }_{x}\right)w,\phantom{\rule{2.em}{0ex}}\left(1\right)$

with $w$ growing not too fast near infinity. In the present paper, we use more ${L}^{2}$-methods and avoid excessive use of the maximum principle. A new observation (at least for the present author) is that the operator in (1) is a conjugated version of a Witten Laplacian in degree 0 and that the differentiated versions of (1) at least in some cases involve Witten Laplacians ${{\Delta }}_{\phi }^{\left(\ell \right)}$ of higher degree $\ell$.

A very natural idea is that in order to get more precise asymptotic results on the correlations, one must analyze the spectrum of the Witten Laplacian in degree 0 a little above the first spectral gap, and the main achievement of the present paper is in making a step in that direction. The main result of the paper describes the asymptotics of the exponential decay of the correlations between ${x}_{j}$ and ${x}_{k}$ as $|j-k|\to \infty$.

##### MSC:
 35P05 General topics in linear spectral theory of PDE 35Q40 PDEs in connection with quantum mechanics 35J10 Schrödinger operator