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 \(|j-k|\to\infty\).

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 |