Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator
Journées équations aux dérivées partielles (2000), article no. 13, 17 p.

Nous poursuivons l’étude de l’opérateur de Kac semiclassique initiée par B. Helffer. Ce type d’opérateurs a été par exemple obtenu par M. Kac lors de l’étude d’un modèle de spins 2D par la méthode dite «de l’opérateur de transfert». On s’intéresse ici à la limite thermodynamique Λ(h) de l’énergie de l’état fondamental de cet opérateur. Pour le modèle de spins de Kac, Λ(h) s’avère être l’énergie libre par spin, et le régime semiclassique correspond à l’approximation de champ moyen. Sous des hypothèses sur le potentiel qui recouvrent les exemples physiques, nous montrons que Λ(h) possède un développement asymptotique formel en puissances de h, et nous en déduisons différentes estimations. Notre approche repose de manière essentielle sur la notion de fonction standard introduite par J. Sjöstrand pour l’étude de questions similaires concernant l’opérateur de Schrödinger. Ceci est un travail en collaboration avec Bernard Helffer.

We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called “transfer operator method”. We are interested here in the thermodynamical limit Λ(h) of the ground state energy of this operator. For Kac’s spin model, Λ(h) is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied by the physical examples we have in mind, we construct a formal asymptotic expansion for Λ(h) in powers of h, from which we derive precise estimates on Λ(h). We work in the settings of Standard Functions introduced by J. Sjöstrand for the study of similar questions in the case of Schrödinger operators.

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     author = {Helffer, Bernard and Ramond, Thierry},
     title = {Semiclassical expansion for the thermodynamic limit of the ground state energy of {Kac's} operator},
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     publisher = {Universit\'e de Nantes},
     year = {2000},
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     mrnumber = {2001i:81087},
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Helffer, Bernard; Ramond, Thierry. Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator. Journées équations aux dérivées partielles (2000), article  no. 13, 17 p. http://www.numdam.org/item/JEDP_2000____A13_0/

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