By Samuel Zaidman

ISBN-10: 0273084275

ISBN-13: 9780273084273

ISBN-10: 0822484277

ISBN-13: 9780822484271

ISBN-10: 1584880112

ISBN-13: 9781584880110

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**Extra resources for Abstract differential equations**

**Example text**

23) implies the Poincare inequality. 20) is sufficient for obtaining the log-Sobolev inequality. 7 Notes • The Witten Laplacians were introduced in a different context by E. Witten [Wit] as an elegant way to prove Morse inequalities on a compact manifold. • The appearance of these Laplacians in the analysis of Poincare inequalities is rather old, at least in Riemannian geometry. In this context, the Witten Laplacians appear as a Bochner Laplacian. '•We shall say that two possibly unbounded selfadjoint operators satisfy the inequality B < A if D(A) C D(B) and if (Bu | u) < (Au | u) , Vu € D(A) .

E . e . Finally if two linearly independent eigenvectors \& and ft corresponding to A exist, we would obtain the same property for ^ by considering as new Hilbert space the orthogonal of ft in W. But it is impossible to have two orthogonal vectors which are strictly positive. 4 Consider the matrix case. 3 in this situation. This is just the standard Perron-Frobenius Theorem. 3 is still valid without the assumption that A is positive. That is show that ||yl|| is actually an eigenvalue of A. 7 Thermodynamic limit Let us first look at the thermodynamic limit.

11) iV—yoo We observe that the speed of convergence to the thermodynamic limit is immediately related to the comparison of A2/A1 with 1. 11). This will explain why we shall later be interested in estimates 27 The 1-d Ising model on the splitting in more general situations. We shall also compute in this very simple situation consider the correlation pair (as • ar+s)N =def Z^1 a ^2 *' CT H-rexp -j3HN{a) . +r)N = Zj1 Yl rKr(T,r')r'KN-r(T',T). 14) where « 3 ( T , T ' ) =TSTIT> , is the so called third Pauli matrix.

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