Out of equilibrium correlations in the XY chain

We study the transversal XY spin-spin correlations in the non-equilibrium steady state constructed in \cite{AP03} and prove their spatial exponential decay close to equilibrium

Non-zero entropy density in the XY chain out of equilibrium

The von Neumann entropy density of a block of n spins is proved to be non-zero for large n in the non-equilibrium steady state of the XY chain constructed by coupling a finite cutout of the chain to the two infinite parts to its left and right which act as thermal reservoirs at different temperatures. Moreover, the non-equilibrium density is shown to be strictly greater than the density in thermal equilibrium

Unique Solutions to Hartree-Fock Equations for Closed Shell Atoms

In this paper we study the problem of uniqueness of solutions to the Hartree and Hartree-Fock equations of atoms. We show, for example, that the Hartree-Fock ground state of a closed shell atom is unique provided the atomic number $Z$ is sufficiently large compared to the number $N$ of electrons. More specifically, a two-electron atom with atomic number $Z\geq 35$ has a unique Hartree-Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some $Z>1$, which exhibits a phase segregation.Comment: 18 page

Long time dynamics and coherent states in nonlinear wave equations

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to appear in Fields Institute Communications: Advances in Applied Mathematics, Modeling, and Computational Science 201