162 research outputs found
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 . 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 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 of
a bridgeless cubic graph , there is a cycle double cover of which
contains . We conjecture that there is even a 5-cycle double cover of
which contains , i.e. is a subgraph of one of the five 2-regular
subgraphs of . We prove a necessary and sufficient condition for a 2-regular
subgraph to be contained in a 5-cycle double cover of
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 is large enough, bipolarons or
multi-polarons will form. When passing through the critical 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
. 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
, while `uncertainty principle' localization energies decay more rapidly,
as .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 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
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