Sjöstrand, J. 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 \(|j-k|\to\infty\). Cited in 3 ReviewsCited in 18 Documents MSC: 35P05 General topics in linear spectral theory for PDEs 35Q40 PDEs in connection with quantum mechanics 35J10 Schrödinger operator, Schrödinger equation Keywords:spectrum of the Witten Laplacian; first spectral gap; exponential decay of the correlations PDF BibTeX XML Cite \textit{J. Sjöstrand}, St. Petersbg. Math. J. 8, No. 1, 160--191 (1996; Zbl 0877.35084); translation from Algebra Anal. 8, No. 1, 160--191 (1996) OpenURL