Approximate Solution of the Representability Problem

Approximate solution of the ensemble representability problem for density operators of arbitrary order is obtained. This solution is closely related to the ``Q condition'' of A.J.Coleman. The representability conditions are formulated in orbital representation and are easy to use. They are tested numerically on the base of CI calculation of simple atomic and molecular systems. General scheme of construction of the contraction operator right inverses is proposed and the explicit expression for the right inverse associated with the expansion operator is derived as an example. Two algorithms for direct 2-density matrix determination are described.Comment: LaTeX2e, 45 pages; significantly revise

On the Lower Garland of Certain Subgroup Lattices in Linear Groups

We describe here the lower garland of some lattices of intermediate subgroups in linear groups. The results are applied to the case of subgroup lattices in general and special linear groups over a class of rings, containing the group of rational points T of a maximal non-split torus in the corresponding algebraic group. It turns out that these garlands coincide with the interval of the whole lattice, consisting of subgroups between T and its normalizer.Comment: AmsTeX, 12 pages; minor correction

(p,q)-Sheaves and the Representability Problem

General properties of new models of the electronic Fock spaces based on the notion of (p,q)-sheaves are studied. Interrelation between simple sheaves and density operators is established. Explicit expressions for the transformed reduced Hamiltonians in terms of the standard creation-annihilation operators are presented. General scheme of parametrization of p-electron states by k-electron means (k=2,3,...) is described and studied in detail for the case of sheaves induced by k-electron wavefunctions. It is demonstrated that under certain conditions p-electron problem may be reformulated as the eigenvalue problem in k-electron space equipped with certain p-electron metric. Simple numerical examples are given to illustrate our approach.Comment: 41 pages, latex, no figures, submitted to IJQ

Intermediate Semigroups are Groups

We consider the lattice of subsemigroups of the general linear group over an Artinian ring containing the group of diagonal matrices and show that every such semigroup is actually a group.Comment: Plain TeX, 6 pages; final version accepted for publication in Semigroup Foru

A Note on the Arrangement of Subgroups in the Automorphism Groups of Submodule Lattices of Free Modules

A complete description of subgroups in the general linear group over a semilocal ring containing the group of diagonal matrices was obtained by Z.I.Borewicz and N.A.Vavilov. It is shown in the present paper that a similar description holds for the intermediate subgroups of the group of all automorphisms of the lattice of right submodules of a free finite rank R-module over a simple Artinian ring containing the group consisting of those automorphisms which leave invariant an appropriate sublattice.Comment: AmsTeX, 6 pages, compile twice; revised version of the Bielefeld preprint 99-00

Galois Theory for a Class of Complete Modular Lattices

We construct Galois theory for sublattices of certain complete modular lattices and their automorphism groups. A well-known description of the intermediate subgroups of the general linear group over a semilocal ring containing the group of diagonal matrices, due to Z.I.Borewicz and N.A.Vavilov, can be obtained as a consequence of this theory.Comment: AmsTeX, 4 pages; Translation into English from Zap. Nauchn. Semin. POMI 236 (1997), 129-132, by A. Pani

Electronic Fock space as associative superalgebra

New algebraic structure on electronic Fock space is studied in detail. This structure is defined in terms of a certain multiplication of many electron wave functions and has close interrelation with coupled cluster and similar approaches. Its study clarifies and simplifies the mathematical backgrounds of these approaches. And even more, it leads to many relations that would be very difficult to derive using conventional technique. Formulas for action of the creation-annihilation operators on products of state vectors are derived. Explicit expressions for action of simplest particle-conserving products of the creation-annihilation operators on powers of state vectors are given. General scheme of parametrization of representable density operators of arbitrary order is presented.Comment: LaTex, 26 pages; one equation added, misprints remove

Pure Representability Problem and New Models of the Electronic Fock Space

New models of the Fock space sector corresponding to some fixed number of electrons are introduced. These models originate from the representability theory and their practical implementation may lead to essential reduction of dimensions of intermediate Configuration Interaction spaces. A certain zero-order theory that gives wave functions approximately equivalent to ones obtained by accounting all excitations from the Hartree-Fock reference state up to the q-th order is proposed. Simple numerical examples are given to illustrate the approach.Comment: 28 pages, submitted to Int. J. Quantum Che

Restoration of Many Electron Wave Functions from One-Electron Density

General theorem describing a relation between diagonal of one-electron density matrix and a certain class of many-electron ensembles of determinant states is proved. As a corollary to this theorem a constructive proof of sufficiency of Coleman's representability conditions is obtained. It is shown that there exist rigorous schemes for construction of energy of many-electron system as functionals of one-electron density.Comment: LaTex, 10 page

On Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes:I

Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k-smooth affine irreducible variety, let K be the fraction field of R, and let G be a reductive simple simply connected R-group scheme isotropic over R. We prove that for any Noetherian k-algebra A, the map of etale cohomology sets H^1(A\otimes_k R,G)-> H^1(A\otimes_ k K,G), induced by the inclusion of R into K, has trivial kernel. This implies the Serre-Grothendieck conjecture for such groups G. The main theorem for A=k and some other results of the present paper are used significantly in arXiv:1211.2678 to prove the Serre-Grothendieck conjecture for all reductive groups over a regular semi-local ring containing an infinite field.Comment: We have incorporated arXiv:1204.1729 and arXiv:0910.5465 into this tex