Semi-classical Limits of the Riesz Means of N-body Schrödinger operators

It is well known that the N-body problem is one of the most important problems in physics and mathematics. The spectrum of a class of N-body Schrödinger operators is considered in this paper. There are two kinds of discrete eigenvalues of these operators, the eigenvalues less than the lower bound of the potential and the eigenvalues larger than the lower bound of the potential. The Dirichlet-Neumann bracket is used to obtain the semi-classical of the number and the Riesz means of the first kind of eigenvalues. The semi-classical estimate of the Riesz means of second kind of eigenvalues can be obtained by the estimate of the difference between the biggest eigenvalue and the lower bound of the potential. We get the estimate of the order of the remainder term of Riesz means in this paper.