*(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 ${e}^{-2\varphi \left(x\right)/h}dx$ on ${\mathbb{R}}^{m}$, where $\varphi $ is a smooth convex function, with special attention to the limit as $m$ tends to infinity. We observed (under suitable assumptions) that the expectation $\langle u\rangle $ of a function $u\left(x\right)$ can be obtained by solving the equation

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 ${{\Delta}}_{\varphi}^{\left(\ell \right)}$ 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 |