Bernard Helffer is a professor at the University Paris-Sud, Paris. He is specializing in semi-classical analysis and its applications in statistical physics. He has published over 150 papers in major journals.

This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform Log–Sobolev inequality.

Contents:

• Witten Laplacians Approach
• Problems in Statistical Mechanics with Discrete Spins
• Laplace Integrals and Transfer Operators
• Semi-Classical Analysis for the Transfer Operators
• Basic Facts in Spectral Theory and on the Schrödinger Operator
• Log-Sobolev Inequalities
• Uniform Decay of Correlations
• Uniform Log-Sobolev Inequalities

"All of the needed material, from spectral theory to semiclassical analysis and statistical mechanics, is given a nice and self-contained presentation as 'one walks through the theory' ... the book is very nicely written, and should be recommended to everyone interested in problems related to partial differential equations and statistical mechanics."

“Written by the hand of a master the book will be useful to all researchers who work in these topics. In conclusion, a wonderful work.”