Helffer, Bernard Semi-classical analysis for the transfer operator: Formal WKB constructions in large dimension. (English) Zbl 0884.35133 Commun. Math. Phys. 187, No. 1, 81-113 (1997). Summary: This paper is devoted to problems coming from statistical mechanics. The transfer matrix (or transfer operator) approach consists in reducing the analysis of asymptotic properties of statistical systems to the analysis of the spectral properties of their transfer operator. Sometimes the new problem appears to have a semiclassical nature. Although the problem is similar to the semiclassical study of Kac’s operator presented in our paper with M. Brunaud which was devoted to the study of \[ \exp- {V\over 2} \cdot \exp h^2 \Delta \cdot \exp- {V\over 2} \] for \(h\) small, new features appear for the model \[ \exp- {V\over 2h} \cdot \exp {h\over a} \Delta \cdot \exp- {V\over 2h}. \] Our first results concern semi-classical analysis of the ground state for this second operator. We then analyze the two models in the large dimension situation. One basic technique is Sjöstrand’s formalism of the 0-standard functions. The one-dimensional case was presented in (1996; Zbl 0847.35114)]. MSC: 35Q40 PDEs in connection with quantum mechanics 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 82C70 Transport processes in time-dependent statistical mechanics Keywords:transfer matrix; spectral properties; Kac’s operator; semi-classical analysis of the ground state; Sjöstrand’s formalism; 0-standard functions Citations:Zbl 0847.35114 PDFBibTeX XMLCite \textit{B. Helffer}, Commun. Math. Phys. 187, No. 1, 81--113 (1997; Zbl 0884.35133) Full Text: DOI