The diamagnetic inequality is established for the Schrödinger operator H (d) 0in L 2 (R d ), d = 2, 3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in R 2 , e.g., a lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schrödinger operator H (d) 0 − V , using new Hardy type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.
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.
Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small o® -diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians.
Abstract. We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N -electron Coulomb systems with quasi-relativistic kinetic energy −α −2 ∆ xn + α −4 −α −2 for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z tot of K nuclei is greater than N − 1 and that Z tot is smaller than a critical charge Z c . The proofs are based on a new application of the Lions-FangGhoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.
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