Densite d'etat surfacique pour une classe d'operateurs de Schrodinger du type a N-corps

Abstract

We are interested in quantum systems composed of a finite number of particles and described by Hamiltonians which are random Schrodinger operators $H^{\omega}:=-\Delta + V^{\omega} $ on $L^2(X)$, where $X$ is a finite dimensional Euclidean space and $\Delta$ is the Laplace-Beltrami operator on $X$. We consider $X$ as the configuration space of the system and we assume that $\{X_n\}_{1\leqslant n \leqslant N_0}$ is a family of linear subspaces of $X$. The orthogonal complement of $X_n$ in $X$ is denoted $X^{n}$ and is considered as the configuration space of a subsystem. We assume that $V^{\omega}$ is a sum of potentials $v_n^{\omega}: X \longrightarrow \R,\quad 1\leqslant n \leqslant N_0,$ which are ergodic with respect the translation group of $X_n$ and which are rapidly decaying in any direction of $X^{n}.$ The aim of this paper is to show the existence of a thermodynamical limit. This limit defines an object which is a type of a the integrated density of states in the case of two body systems.