The Gaussian core model in high dimensions

We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t \mapsto e^{-\alpha t^2}$ with $0 < \alpha < 4\pi/e$, we show that no point configuration in $\mathbf{R}^n$ of density $\rho$ can have energy less than $(\rho+o(1))(\pi/\alpha)^{n/2}$ as $n \to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho (\pi/\alpha)^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling-Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi/\alpha)^{n/2}$ is no longer asymptotically sharp when $\alpha > \pi e$. As a consequence of our results, we obtain bounds in $\mathbf{R}^n$ for the minimal energy under inverse power laws $t \mapsto 1/t^{n+s}$ with $s>0$, and these bounds are sharp to within a constant factor as $n \to \infty$ with $s$ fixed.Comment: 30 pages, 1 figur

A Sharp Bound on the ss-Energy and Its Applications to Averaging Systems

The {\em $s$-energy} is a generating function of wide applicability in network-based dynamics. We derive an (essentially) optimal bound of $(3/\rho s)^{n-1}$ on the $s$-energy of an $n$-agent symmetric averaging system, for any positive real $s\leq 1$, where~$\rho$ is a lower bound on the nonzero weights. This is done by introducing the new dynamics of {\em twist systems}. We show how to use the new bound on the $s$-energy to tighten the convergence rate of systems in opinion dynamics, flocking, and synchronization

Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

This article provides sharp constructive upper and lower bound estimates for the non-linear Boltzmann collision operator with the full range of physical non cut-off collision kernels ($\gamma > -n$ and $s\in (0,1)$) in the trilinear $L^2(\R^n)$ energy $$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [2009, 2010, 2010 arXiv:1011.5441v1]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(\R^n)$.Comment: 29 pages, updated file based on referee report; Advances in Mathematics (2011

Moment methods in energy minimization: New bounds for Riesz minimal energy problems

We use moment methods to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate the infinite dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We compute the second step of our hierarchy for Riesz $s$-energy problems with five particles on the $2$-dimensional unit sphere, where the $s=1$ case known as the Thomson problem. This yields new sharp bounds (up to high precision) and suggests the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for $5$-particles on $S^2$. This is the first time a $4$-point bound has been computed for a continuous problem

A balancing act: Evidence for a strong subdominant d-wave pairing channel in Ba0.6K0.4Fe2As2{\rm Ba_{0.6}K_{0.4}Fe_2As_2}

We present an analysis of the Raman spectra of optimally doped ${\rm Ba_{0.6}K_{0.4}Fe_2As_2}$ based on LDA band structure calculations and the subsequent estimation of effective Raman vertices. Experimentally a narrow, emergent mode appears in the $B_{1g}$ ($d_{x^2-y^2}$) Raman spectra only below $T_c$, well into the superconducting state and at an energy below twice the energy gap on the electron Fermi surface sheets. The Raman spectra can be reproduced quantitatively with estimates for the magnitude and momentum space structure of the s$_{+-}$ pairing gap on different Fermi surface sheets, as well as the identification of the emergent sharp feature as a Bardasis-Schrieffer exciton, formed as a Cooper pair bound state in a subdominant $d_{x^2-y^2}$ channel. The binding energy of the exciton relative to the gap edge shows that the coupling strength in this subdominant $d_{x^2-y^2}$ channel is as strong as 60% of that in the dominant $s_{+-}$ channel. This result suggests that $d_{x^2-y^2}$ may be the dominant pairing symmetry in Fe-based sperconductors which lack central hole bands.Comment: 10 pages, 6 Figure

Geodesics and horizontal-path spaces in Carnot groups

We study the topology of horizontal-paths spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the paths space). These geodesics typically appear in families (critical manifolds). Letting the energy grow, we obtain an upper bound on the number of critical manifolds with energy bounded by s: this upper bound is polynomial in s of degree l (the corank of the distribution). Despite this evidence, we show that Morse-Bott inequalities are far from sharp: the topology (i.e. the sum of the Betti numbers) of the loop space filtered by the energy grows at most as a polynomial in s of degree l-1. In the limit for s at infinity, all Betti numbers (except the zeroth) must actually vanish: the admissible-loop space is contractible. In the case the corank l=2 we compute exactly the leading coefficient of the sum of the Betti numbers of the admissible-loop space with energy less than s. This coefficient is expressed by an integral on the unit circle depending only on the coordinates of the final point and the structure constants of the Lie algebra of G

Intrinsic excitonic photoluminescence and band-gap engineering of wide-gap p-type oxychalcogenide epitaxial films of LnCuOCh (Ln = La, Pr, and Nd; Ch = S or Se) semiconductor alloys

The optical spectroscopic properties of layered oxychalcogenide semiconductors LnCuOCh (Ln = La, Pr, and Nd; Ch = S or Se) on epitaxial films were thoroughly investigated near the fundamental energy band edges. Free exciton emissions were observed for all the films between 300 and ~30 K. In addition, a sharp emission line, which was attributed to bound excitons, appeared below ~80 K. The free exciton energy showed a nonmonotonic relationship with lattice constant and was dependent on lanthanide and chalcogen ion substitutions. These results imply that the exciton was confined to the (Cu2Ch2)2– layer. Anionic and cationic substitutions tune the emission energy at 300 K from 3.21 to 2.89 eV and provide a way to engineer the electronic structure in light-emitting devices