410 research outputs found
Electron Wavefunctions and Densities for Atoms
With a special `Ansatz' we analyse the regularity properties of atomic
electron wavefunctions and electron densities. In particular we prove an a
priori estimate, and obtain for the spherically averaged electron density,
, that exists and is non-negative
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
Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei
We investigate regularity properties of molecular one-electron densities rho
near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x))
with an explicit function F, only depending on the nuclear charges and the
positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has
locally essentially bounded second derivatives. An example constructed using
Hydrogenic eigenfunctions shows that this regularity result is sharp. For
atomic eigenfunctions which are either even or odd with respect to inversion in
the origin, we prove that mu is even C^{2,\alpha}(R^3) for all alpha in (0,1).
Placing one nucleus at the origin we study rho in polar coordinates x=r*omega
and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to
zero. We prove non-isotropic cusp conditions of first and second order, which
generalize Kato's classical result.Comment: 19 page
Positivity and lower bounds to the decay of the atomic one-electron density
We investigate properties of the spherically averaged atomic one-electron
density rho~(r). For a rho~ which stems from a physical ground state we prove
that rho~ > 0. We also give exponentially decreasing lower bounds to rho~ in
the case when the eigenvalue is below the corresponding essential spectrum.Comment: 20 page
Nash Williams Conjecture and the Dominating Cycle Conjecture
The disproved Nash Williams conjecture states that every 4-regular
4-connected graph has a hamiltonian cycle. We show that a modification of this
conjecture is equivalent to the Dominating Cycle Conjecture
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
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