589 research outputs found
The electron densities of pseudorelativistic eigenfunctions are smooth away from the nuclei
We consider a pseudorelativistic model of atoms and molecules, where the
kinetic energy of the electrons is given by . In this model
the eigenfunctions are generally not even bounded, however, we prove that the
corresponding one-electron densities are smooth away from the nuclei.Comment: 16 page
Source Coding in Networks with Covariance Distortion Constraints
We consider a source coding problem with a network scenario in mind, and
formulate it as a remote vector Gaussian Wyner-Ziv problem under covariance
matrix distortions. We define a notion of minimum for two positive-definite
matrices based on which we derive an explicit formula for the rate-distortion
function (RDF). We then study the special cases and applications of this
result. We show that two well-studied source coding problems, i.e. remote
vector Gaussian Wyner-Ziv problems with mean-squared error and mutual
information constraints are in fact special cases of our results. Finally, we
apply our results to a joint source coding and denoising problem. We consider a
network with a centralized topology and a given weighted sum-rate constraint,
where the received signals at the center are to be fused to maximize the output
SNR while enforcing no linear distortion. We show that one can design the
distortion matrices at the nodes in order to maximize the output SNR at the
fusion center. We thereby bridge between denoising and source coding within
this setup
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
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