We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.2010 Mathematics Subject Classification. Primary: 53Z05; Secondary: 81V55,22E65, 58B20.
Abstract. Let I be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and U(H) I the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. In this paper we study the geometry of the unitary orbitswhere V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + I, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V * V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + I and homogeneous reductive spaces of U(H) I and U(H) I × U(H) I respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(H) I (or U(H) I × U(H) I ) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.
Let Ω be an open subset of R n . Let L 2 = L 2 (Ω, dx) and H 1 0 = H 1 0 (Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H 1 0 which preserve the L 2 -inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H 1 0 solutions of the non-homogeneous Helmholtz equation u − ∆u = f , u| ∂Ω = 0. We show that G is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups G p := G ∩ (I − B p (H 1 0 )), where B p (H 1 0 ) is a Schatten ideal of operators on H 1 0 . An invariant (weak) Finsler metric is defined by the p-norm of the Schatten ideal of operators of L 2 . We prove that any pair of operators G 1 , G 2 ∈ G p can be joined by a minimal curve of the form δ(t) = G 1 e itX , where X is a symmetrizable operator in B p (H 1 0 ).
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