Existence of ground states for a modified nonlinear Schrodinger equation

In this paper we prove existence of ground state solutions of the modified nonlinear Schrodinger equation: $$-\Delta u+V(x)u-{1/2}u \Delta u^{2}=|u|^{p-1}u, x \in \R^N, N \geq 3,$$ under some hypotheses on $V(x)$. This model has been proposed in the theory of superfluid films in plasma physics. As a main novelty with respect to some previous results, we are able to deal with exponents $p\in(1,3)$. The proof is accomplished by minimization under a convenient constraint

Fluctuational Electrodynamics in Atomic and Macroscopic Systems: van der Waals Interactions and Radiative Heat Transfer

We present an approach to describing fluctuational electrodynamic (FED) interactions, particularly van der Waals (vdW) interactions as well as radiative heat transfer (RHT), between material bodies of vastly different length scales, allowing for going between atomistic and continuum treatments of the response of each of these bodies as desired. Any local continuum description of electromagnetic (EM) response is compatible with our approach, while atomistic descriptions in our approach are based on effective electronic and nuclear oscillator degrees of freedom, encapsulating dissipation, short-range electronic correlations, and collective nuclear vibrations (phonons). While our previous works using this approach have focused on presenting novel results, this work focuses on the derivations underlying these methods. First, we show how the distinction between "atomic" and "macroscopic" bodies is ultimately somewhat arbitrary, as formulas for vdW free energies and RHT look very similar regardless of how the distinction is drawn. Next, we demonstrate that the atomistic description of material response in our approach yields EM interaction matrix elements which are expressed in terms of analytical formulas for compact bodies or semianalytical formulas based on Ewald summation for periodic media; we use this to compute vdW interaction free energies as well as RHT powers among small biological molecules in the presence of a metallic plate as well as between parallel graphene sheets in vacuum, showing strong deviations from conventional macroscopic theories due to the confluence of geometry, phonons, and EM retardation effects. Finally, we propose formulas for efficient computation of FED interactions among material bodies in which those that are treated atomistically as well as those treated through continuum methods may have arbitrary shapes, extending previous surface-integral techniques.Comment: 25 pages, 5 figures, 2 appendice