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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 -2ϕ(x)/h dx on m , where ϕ 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(x) can be obtained by solving the equation

u-u=(-h 2 Δ+2ϕ·h x )w,(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 Δ ϕ () of higher degree .

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|.


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