INTEGRAL-GEOMETRICAL CONSIDERATION OF DENSITY MATRICES

摘要

The ensemble N-representability problem for the k-th order reduced density matrix (k-RDM) as well as the problem of reconstruction of the N-particle system density matrices (N-DM) from a given k-RDM are studied. The spatial parts of the k-RDM expansion in terms of spin tensorial operators Theta(lambda)(k) are represented using particular values (at specially chosen Xi = Xi(0)) of the Radon transform D-N lambda(Xi) of the N-DM spatial parts (or their sums) D-N lambda(x'x '') (here, Xi is a d-plane in the n-space R(n) of x = (x', x ''), with n = 6N, d = 3 (N - k), x' = (r'(1),...,r'(N)), x '' = (r ''(1),...,r ''(N))). In this way, the problem is reduced to investigation of the properties of the functions D-N lambda(Xi). For a normalizable N - DM, it is proved that D-N lambda(Xi) are bounded functions. The properties of D(N lambda)Xi implied by the N-DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N-DM corresponding to a given k-RDM is proposed. (C) 1995 John Wiley & Sons, Inc. [References: 41]