Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $\mathcal{O}(\varepsilon)$ of $u^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension $n = 1$.Comment: 22 pages, typos corrected, references added, final accepted versio

Clustering of vacancy defects in high-purity semi-insulating SiC

Positron lifetime spectroscopy was used to study native vacancy defects in semi-insulating silicon carbide. The material is shown to contain (i) vacancy clusters consisting of 4--5 missing atoms and (ii) Si vacancy related negatively charged defects. The total open volume bound to the clusters anticorrelates with the electrical resistivity both in as-grown and annealed material. Our results suggest that Si vacancy related complexes compensate electrically the as-grown material, but migrate to increase the size of the clusters during annealing, leading to loss of resistivity.Comment: 8 pages, 5 figure

Vanishing discount problems for Hamilton--Jacobi equations on changing domains

We study the asymptotic behavior, as $\lambda\rightarrow 0^+$, of the Hamilton-Jacobi equation $\phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) = 0$ in $(1+r(\lambda))\Omega$ with state-constraint boundary condition. Here, $\Omega$ is a bounded domain of $\mathbb{R}^n$, $\phi(\lambda), r(\lambda):(0,\infty)\rightarrow \mathbb{R}$ are continuous functions such that $\phi$ is nonnegative and $\lim_{\lambda\rightarrow 0^+} \phi(\lambda) = \lim_{\lambda\rightarrow 0^+} r(\lambda) = 0$. Surprisingly, we are able to obtain both convergence results and non-convergence results in this convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of $H$ in $(1+r(\lambda))\Omega$ as $\lambda\rightarrow 0^+$. The main tool we use is a duality representation of solution with viscosity Mather measures.Comment: 31 pages, 3 figures. AMSart style, errors fixe

Universal Properties of Two-Dimensional Boson Droplets

We consider a system of N nonrelativistic bosons in two dimensions, interacting weakly via a short-range attractive potential. We show that for N large, but below some critical value, the properties of the N-boson bound state are universal. In particular, the ratio of the binding energies of (N+1)- and N-boson systems, B_{N+1}/B_N, approaches a finite limit, approximately 8.567, at large N. We also confirm previous results that the three-body system has exactly two bound states. We find for the ground state B_3^(0) = 16.522688(1) B_2 and for the excited state B_3^(1) = 1.2704091(1) B_2.Comment: 4 pages, 2 figures, final versio

Spontaneous Symmetry Breaking with Abnormal Number of Nambu-Goldstone Bosons and Kaon Condensate

We describe a class of relativistic models incorporating finite density of matter in which spontaneous breakdown of continuous symmetries leads to a lesser number of Nambu-Goldstone bosons than that required by the Goldstone theorem. This class, in particular, describes the dynamics of the kaon condensate in the color-flavor locked phase of high density QCD. We describe the spectrum of low energy excitations in this dynamics and show that, despite the presence of a condensate and gapless excitations, this system is not a superfluid.Comment: 5 pages, 1 figure, REVTeX. Minor revisions made and 2 new references added. To appear in Phys. Rev. Let

Inverse meson mass ordering in color-flavor-locking phase of high density QCD: erratum

We correct a mistake in the calculation of meson masses at large baryon chemical potential made in hep-ph/9910491v2Comment: 2 pages, 1 figure, erratum to hep-ph/9910491v

Characterization of the nitrogen split interstitial defect in wurtzite aluminum nitride using density functional theory

We carried out Heyd-Scuseria-Ernzerhof hybrid density functional theory plane wave supercell calculations in wurtzite aluminum nitride in order to characterize the geometry, formation energies, transition levels and hyperfine tensors of the nitrogen split interstitial defect. The calculated hyperfine tensors may provide useful fingerprint of this defect for electron paramagnetic resonance measurement.Comment: 5 pages, 3 figure