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.     

Semiclassics of the quantum current

We give the semi–classical asymptotics of the quntum current in 2 cases: First fro T>0 (T is the absolute temperatur). Here we get a complete asymptotices. Then for T=0. Here it vanishes to the accessible orders.     

On the Third Critical Field in Ginzburg-Landau Theory

Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, , describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables.The two authors are supported by the European Research Network ‘Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems’ with contract number HPRN-CT-2002-00277, and the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). Part of this work was carried out while S.F. visited CIMAT, Mexico.     

An optimal semiclassical bound on commutators of spectral projections with position and momentum operators

Letters in Mathematical Physics - We prove an optimal semiclassical bound on the trace norm of the following commutators $$[{\varvec{1}}_{(-\infty ,0]}(H_\hbar ),x]$$ , $$[{\varvec{1}}_{(-\infty...     

Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei

We investigate regularity properties of molecular one-electron densities ρ near the nuclei. In particular we derive a representation

Inequalities for the lowest magnetic Neumann eigenvalue

We study the ground-state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field, we consider the question whether the disc maximizes this eigenvalue for fixed area. More generally, we discuss old and new bounds obtained on this problem.

     

Energy asymptotics for Type II superconductors

We study the Ginzburg–Landau functional in the parameter regime describing ‘Type II superconductors’. In the exact regime considered minimizers are localized to the boundary — i.e. the sample is only superconducting in the boundary region. Depending on the relative size of different parameters we describe the concentration behavior and give leading order energy asymptotics. This generalizes previous results by Lu–Pan, Helffer–Pan, and Pan.     

Third Derivative of the One-Electron Density at the Nucleus

We study electron densities of eigenfunctions of atomic Schr?dinger operators. We prove the existence of , the third derivative of the spherically averaged atomic density at the nucleus. For eigenfunctions with corresponding eigenvalue below the essential spectrum in any symmetry subspace we obtain the bound , where Z denotes the nuclear charge. This bound is optimal. ? 2008 by the authors. This article may be reproduced in its entirety for non-commercial purposes. Submitted: April 22, 2008. Accepted: July 7, 2008.