# Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator

Bernard Helffer; Thierry Ramond

Journées équations aux dérivées partielles (2000)

- page 1-17
- ISSN: 0752-0360

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topHelffer, Bernard, and Ramond, Thierry. "Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator." Journées équations aux dérivées partielles (2000): 1-17. <http://eudml.org/doc/93390>.

@article{Helffer2000,

abstract = {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 $\Lambda (h)$ of the ground state energy of this operator. For Kac’s spin model, $\Lambda (h)$ 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 $\Lambda (h)$ in powers of $h$, from which we derive precise estimates on $\Lambda (h)$. We work in the settings of Standard Functions introduced by J. Sjöstrand for the study of similar questions in the case of Schrödinger operators.},

author = {Helffer, Bernard, Ramond, Thierry},

journal = {Journées équations aux dérivées partielles},

keywords = {transfer operator; Kac operator; thermodynamical limit; standard functions},

language = {eng},

pages = {1-17},

publisher = {Université de Nantes},

title = {Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator},

url = {http://eudml.org/doc/93390},

year = {2000},

}

TY - JOUR

AU - Helffer, Bernard

AU - Ramond, Thierry

TI - Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator

JO - Journées équations aux dérivées partielles

PY - 2000

PB - Université de Nantes

SP - 1

EP - 17

AB - 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 $\Lambda (h)$ of the ground state energy of this operator. For Kac’s spin model, $\Lambda (h)$ 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 $\Lambda (h)$ in powers of $h$, from which we derive precise estimates on $\Lambda (h)$. We work in the settings of Standard Functions introduced by J. Sjöstrand for the study of similar questions in the case of Schrödinger operators.

LA - eng

KW - transfer operator; Kac operator; thermodynamical limit; standard functions

UR - http://eudml.org/doc/93390

ER -

## References

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- [Sj1] J. Sjöstrand : Potential wells in high dimensions I, Ann. de l'Institut Henri Poincaré, Phys. Théor., Vol. 58 n.1 (1993), p. 1-41. Zbl0770.35050MR94k:81055a
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- [Sj3] J. Sjöstrand : Evolution equations in a large number of variables, Math. Nachr. 166 (1994), p. 17-53. Zbl0837.35061MR95h:35094

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