6533b822fe1ef96bd127d5c1

RESEARCH PRODUCT

Integral-geometrical consideration of density matrices

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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}{sub {lambda}} are represented using particular values (at specially chosen {Xi} = {Xi}{sub o}) of the Radon transform D{sub N{lambda}} D{sub N{lambda}}({Xi}) of the N-DM spatial parts (or their sums) D{sub N{lambda}}({chi}{prime}{vert_bar}{chi}{double_prime}) (here, {Xi} is a d-plane in the n-space {Re}{double_prime} of {chi} = ({chi}{prime}, {chi}{double_prime}), with n = 6N, d = 3(N - k), {chi}{prime} {double_bond}{prime} (r{prime}{sub 1},...,r{sub N}{prime}), {chi}{double_prime} {double_prime} {triple_bond} (r{sub 1}{double_prime},...,r{sub N}{double_prime})). In this way, the problem is reduced to investigation of the properties of the functions D{sub N{lambda}}({Xi}). For a normalizable N - DM, it is proved that D{sub N{lambda}}({Xi}) are bounded functions. The properties of D{sub 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. 38 refs.