## Correlation asymptotics and Witten Laplacians.(English)Zbl 0877.35084

St. Petersbg. Math. J. 8, No. 1, 123-147 (1997); translation from Algebra Anal. 8, No. 1, 160-191 (1996).
In an earlier work [J. Stat. Phys. 74, 349-409 (1994)], B. Helffer and the author studied correlations associated with the measure $$e^{-2\phi(x)/h}dx$$ on $$\mathbb{R}^m$$, 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 $$\langle u\rangle$$ of a function $$u(x)$$ can be obtained by solving the equation $u-\langle u\rangle= (-h^2\Delta+ 2\nabla\phi\cdot h\partial_x)w,\tag{1}$ 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 $$\Delta^{(\ell)}_\phi$$ 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 for PDEs 35Q40 PDEs in connection with quantum mechanics 35J10 Schrödinger operator, Schrödinger equation