We construct states on a C★-algebra associated to a one dimensional lattice crystal. We also compute the mean value of an observable, not necessarily bounded, such as the dilation coefficient. This implies on one hand, a careful analysis of the heat kernel of the Hamiltonian associated to the crystal and, on the other hand, the study of the quantum correlations of two observables associated to two clusters of particules.
Nous construisons des états d'équilibre sur une C★ algèbre associée à un cristal quantique unidimensionnel. Nous étudions la valeur moyenne d'une observable, non nécessairement bornée, telle que le coefficient de dilatation. Ceci demande, d'une part, une analyse précise du noyau de la chaleur associé au cristal et, d'autre part, l'étude des corrélations quantiques de deux observables associés a deux amas de particules.
Accepted:
Published online:
Laurent Amour 1; Claudy Cancelier 1; Pierre Levy-Bruhl 1; Jean Nourrigat 1
@article{CRMATH_2003__336_12_981_0, author = {Laurent Amour and Claudy Cancelier and Pierre Levy-Bruhl and Jean Nourrigat}, title = {States of a one dimensional quantum crystal}, journal = {Comptes Rendus. Math\'ematique}, pages = {981--984}, publisher = {Elsevier}, volume = {336}, number = {12}, year = {2003}, doi = {10.1016/S1631-073X(03)00229-2}, language = {en}, }
TY - JOUR AU - Laurent Amour AU - Claudy Cancelier AU - Pierre Levy-Bruhl AU - Jean Nourrigat TI - States of a one dimensional quantum crystal JO - Comptes Rendus. Mathématique PY - 2003 SP - 981 EP - 984 VL - 336 IS - 12 PB - Elsevier DO - 10.1016/S1631-073X(03)00229-2 LA - en ID - CRMATH_2003__336_12_981_0 ER -
Laurent Amour; Claudy Cancelier; Pierre Levy-Bruhl; Jean Nourrigat. States of a one dimensional quantum crystal. Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 981-984. doi : 10.1016/S1631-073X(03)00229-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00229-2/
[1] Euclidean Gibbs states of quantum crystals, Moscow Math. J., Volume 1 (2001) no. 3, pp. 307-313
[2] Gibbs states on loop lattices: existence and a priori estimates, C. R. Acad. Sci. Paris, Sér. I, Volume 333 (2001), pp. 1005-1009
[3] L. Amour, C. Cancelier, P. Levy-Bruhl, J. Nourrigat, The heat kernel for Hamiltonians on lattices crystals, Prépublication 03.01, Reims, 2003
[4] Solid State Physics, Saunders College, Fort Worth, 1976
[5] B. Helffer, Semi-classical analysis for Schrödinger operators, Laplace integrals and transfer operators in large dimension: an introduction, Cours, Université de Paris-Sud, 1995
[6] Remarks on the decay of correlations and Witten Laplacians, Brascamp–Lieb inequalities and semi-classical limit, J. Funct. Anal., Volume 155 (1998) no. 2, pp. 571-586
[7] Remarks on the decay of correlations and Witten Laplacians, II. Analysis of the dependence of the interaction, Rev. Math. Phys., Volume 11 (1999) no. 3, pp. 321-336
[8] Remarks on the decay of correlations and Witten Laplacians, III. Applications to the logarithmic Sobolev inequalities, Ann. Inst. H. Poincaré Probab. Statist., Volume 35 (1999) no. 4, pp. 483-508
[9] Semiclassical expansions of the thermodynamic limit for a Schrödinger equation. I. The one well case, Méthodes semi-classiques, Astérique 210, 2, Soc. Math. France, Paris, 1992
[10] On the correlation for Kac like models in the convex case, J. Statist. Phys., Volume 74 (1994) no. 1,2, pp. 349-409
[11] Introduction to Solid State Physics, Wiley, New York, 1976
[12] Analyticity of the Gibbs states for a quantum anharmonic crystal: no order parameter, Ann. Inst. H. Poincaré, Volume 3 (2002), pp. 921-938
[13] A quantum crystal model in the light-mass limit: Gibbs states, Rev. Math. Phys., Volume 12 (2000) no. 7, pp. 981-1032
[14] Ch. Royer, Formes quadratiques et calcul pseudodifférentiel en grande dimension, Prépublication 00.05, Reims, 2000
[15] The Statistical Mechanics of Lattice Gases, Princeton Ser. in Phys., I, Princeton University Press, Princeton, 1993
[16] Evolution equations in a large number of variables, Math. Nachr., Volume 166 (1994), pp. 17-53
[17] Complete asymptotics for correlations of Laplace integrals in the semiclassical limit, Mém. Soc. Math. France, Volume 83 (2000)
Cited by Sources:
Comments - Policy