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Correlation asymptotics and Witten Laplacians. (English) Zbl 0877.35084
In an earlier work [J. Stat. Phys. 74, 349-409 (1994)], {\it B. Helffer} and the author studied correlations associated with the measure $e^{-2\phi(x)/h}dx$ on $\bbfR^m$, 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 $$u-\langle u\rangle= (-h^2\Delta+ 2\nabla\phi\cdot h\partial_x)w,\tag1$$ 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 $|j-k|\to\infty$.

35P05General topics in linear spectral theory of PDE
35Q40PDEs in connection with quantum mechanics
35J10Schrödinger operator