Many Particle Hardy-Inequalities

In this paper we prove three differenttypes of the so-called many-particle Hardy inequalities. One of them is a "classical type" which is valid in any dimesnion $d\neq 2$. The second type deals with two-dimensional magnetic Dirichlet forms where every particle is supplied with a soplenoid. Finally we show that Hardy inequalities for Fermions hold true in all dimensions.Comment: 20 page

Analyticity of the density of electronic wavefunctions

We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in ${\mathbb R}^3$ away from the nuclei.Comment: 19 page

ANALYTIC STRUCTURE OF SOLUTIONS TO MULTICONFIGURATION EQUATIONS

Abstract. We study the regularity at the positions of the (fixed) nuclei of solutions to (non-relativistic) multiconfiguration equations (including Hartree–Fock) of Coulomb systems. We prove the following: Let {ϕ1,..., ϕM} be any solution to the rank–M multiconfiguration equations for a molecule with L fixed nuclei at R1,..., RL ∈ R 3. Then, for any j ∈ {1,..., M}, k ∈ {1,..., L}, there exists a neighbourhood Uj,k ⊆ R 3 of Rk, and functions ϕ (1) j,k, ϕ(2) j,k, real analytic in Uj,k, such that ϕj(x) = ϕ (1) (2) j,k (x) + |x − Rk|ϕ j,k (x), x ∈ Uj,k. A similar result holds for the corresponding electron density. The proof uses the Kustaanheimo–Stiefel transformation, as applied in [9] to the study of the eigenfunctions of the Schrödinger operator of atoms and molecules near two-particle coalescence points. 1. Introduction an

The electron density is smooth away from the nuclei

We prove that the electron densities of electronic eigenfunctions of atoms and molecules are smooth away from the nuclei.Comment: 16 page

A note on 5-cycle double covers

The strong cycle double cover conjecture states that for every circuit $C$ of a bridgeless cubic graph $G$, there is a cycle double cover of $G$ which contains $C$. We conjecture that there is even a 5-cycle double cover $S$ of $G$ which contains $C$, i.e. $C$ is a subgraph of one of the five 2-regular subgraphs of $S$. We prove a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of $G$

Analytic structure of solutions to multiconfiguration equations

We study the regularity at the positions of the (fixed) nuclei of solutions to (non-relativistic) multiconfiguration equations (including Hartree--Fock) of Coulomb systems. We prove the following: Let {phi_1,...,phi_M} be any solution to the rank--M multiconfiguration equations for a molecule with L fixed nuclei at R_1,...,R_L in R^3. Then, for any j in {1,...,M} and k in {1,...,L}, there exists a neighbourhood U_{j,k} in R^3 of R_k, and functions phi^{(1)}_{j,k}, phi^{(2)}_{j,k}, real analytic in U_{j,k}, such that phi_j(x) = phi^{(1)}_{j,k}(x) + |x - R_k| phi^{(2)}_{j,k}(x), x in U_{j,k} A similar result holds for the corresponding electron density. The proof uses the Kustaanheimo--Stiefel transformation, as applied earlier by the authors to the study of the eigenfunctions of the Schr"odinger operator of atoms and molecules near two-particle coalescence points.Comment: 15 page

Binding of Polarons and Atoms at Threshold

If the polaron coupling constant $\alpha$ is large enough, bipolarons or multi-polarons will form. When passing through the critical $\alpha_c$ from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explodes? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at $\alpha_c$. Similarly, we show that the same phenomenon occurs for atoms, e.g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schr\"odinger equation, and are very general. They use the fact that the Coulomb repulsion decays like $1/r$, while `uncertainty principle' localization energies decay more rapidly, as $1/r^2$.Comment: 19 page

Rigorous conditions for the existence of bound states at the threshold in the two-particle case

In the framework of non-relativistic quantum mechanics and with the help of the Greens functions formalism we study the behavior of weakly bound states as they approach the continuum threshold. Through estimating the Green's function for positive potentials we derive rigorously the upper bound on the wave function, which helps to control its falloff. In particular, we prove that for potentials whose repulsive part decays slower than $1/r^{2}$ the bound states approaching the threshold do not spread and eventually become bound states at the threshold. This means that such systems never reach supersizes, which would extend far beyond the effective range of attraction. The method presented here is applicable in the many--body case