Pseudodifferential calculus in large dimension and thermodynamic limits

If P-n(h) is a h-pseudodifferential operator in R-n associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space. with bounded derivatives, we describe the symbol of e(-tPn(h)), by inegalities where the constants depend on the bounds for the derivatives of the symbol of P-n(h), but not on the dimension n. Some applications to thermodynamic limits are given. (C) 2003 Academie des sciences/Editions scientifiques et medicales Elsevier SAS. All rights reserved.