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Ferromagnetic integrals, correlations and maximum principles. (English) Zbl 0831.35031
In a previous paper [Astérisque, 210, 303-326 (1992; Zbl 0796.35123)] the author developed an idea of I. M. Singer, B. Wong, S.-T. Yau and S. S.-T. Yau [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 319-333 (1985; Zbl 0603.35070)] to use the maximum principle in the study of the logarithm of the first eigenfunction of a Schrödinger operator on $$\mathbb{R}^m$$ with a strictly convex potential, and for suitable sequences of potentials, he was able to establish the exponential convergence of the first eigenvalue divided by $$m$$, when $$m$$ tends to infinity. More recently, the reviewer and the author [J. Statistical Phys. 74, No. 1-2, 349-369 (1994)] employed a similar method to study expectation values of the form: $\langle f \rangle = \left. \left( \int_{\mathbb{R}^m} \exp - {\varphi (x) \over h} f(x) dx \right) \right/ \int_{\mathbb{R}^m} \exp - {\varphi (x) \over h} dx,$ and in particular the correlations $$({\mathcal C})$$ $$\langle (x_j - \langle x_j \rangle) (x_k - \langle x_k \rangle) \rangle$$, for large $$m$$ and $$|j - k |$$. Under suitable assumptions, implying uniform strict convexity for $$\varphi$$, it was proved that $$({\mathcal C})$$ can be estimated by $${\mathcal O} (1) \exp - |j - k |/C$$. In this paper, the author improves the use of the maximum principle and as an application considers for the correlations $$({\mathcal C})$$ a critical case and proves power decay upper bounds in terms of the fundamental solution of a certain elliptic operator. In the last part, the author formulates a general maximum principle and gives two applications.
Reviewer: B.Helffer (Orsay)

##### MSC:
 35C20 Asymptotic expansions of solutions to PDEs 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35P15 Estimates of eigenvalues in context of PDEs
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##### References:
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