Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II

We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying \textit{a posteriori} a separate work on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has changed nam

The Isgur-Wise function in a relativistic model for qQˉq\bar Q system

We use the Dirac equation with a ``(asymptotically free) Coulomb + (Lorentz scalar) linear '' potential to estimate the light quark wavefunction for $q\bar Q$ mesons in the limit $m_Q\to \infty$. We use these wavefunctions to calculate the Isgur-Wise function $\xi (v.v^\prime )$ for orbital and radial ground states in the phenomenologically interesting range $1\leq v.v^ \prime \leq 4$. We find a simple expression for the zero-recoil slope, $\xi^ \prime (1) =-1/2- \epsilon^2 /3$, where $\epsilon$ is the energy eigenvalue of the light quark, which can be identified with the $\bar\Lambda$ parameter of the Heavy Quark Effective Theory. This result implies an upper bound of $-1/2$ for the slope $\xi^\prime (1)$. Also, because for a very light quark $q (q=u, d)$ the size $\sqrt {}$ of the meson is determined mainly by the ``confining'' term in the potential $(\gamma_\circ \sigma r)$, the shape of $\xi_{u,d}(v.v^\prime )$ is seen to be mostly sensitive to the dimensionless ratio $\bar \Lambda_{u,d}^2/\sigma$. We present results for the ranges of parameters $150 MeV <\bar \Lambda_{u,d} <600 MeV$ $(\bar\Lambda_s \approx \bar\Lambda_{u,d}+100 MeV)$, $0.14 {GeV}^2 \leq \sigma \leq 0.25 {GeV}^2$ and light quark masses $m_u, m_d \approx 0, m_s=175 MeV$ and compare to existing experimental data and other theoretical estimates. Fits to the data give: ${\bar\Lambda_{u,d}}^2/\sigma =4.8\pm 1.7$, $-\xi^\prime_{u,d}(1)=2.4\pm 0.7$ and $\vert V_{cb} \vert \sqrt {\frac {\tau_B}{1.48 ps}}=0.050\pm 0.008$ [ARGUS '93]; ${\bar\Lambda_{u,d}}^2/\sigma = 3.4\pm 1.8$, $-\xi^\prime_{u,d}(1)=1.8\pm 0.7$ and $\vert V_{cb} \vert \sqrt { \frac {\tau_B}{1.48 ps}}=0.043\pm 0.008$ [CLEO '93]; ${\bar\Lambda_{u,d}}^2/Comment: 22 pages, Latex, 4 figures (not included) available by fax or via email upon reques

Curvature of the universe and the dark energy potential

The flatness of an accelerating universe model (characterized by a dark energy scalar field $\chi$) is mimicked from a curved model that is filled with, apart from the cold dark matter component, a quintessencelike scalar field $Q$. In this process, we characterize the original scalar potential $V(Q)$ and the mimicked scalar potential $V(\chi)$ associated to the scalar fields $Q$ and $\chi$, respectively. The parameters of the original model are fixed through the mimicked quantities that we relate to the present astronomical data, such that the equation state parameter $w_{_{\chi}}$ and the dark energy density parameter $\Omega_{\chi}$.Comment: References 7 and 8 have been corrected: (7) Riess et al. 1998, AJ, 116, 1009 and (8) Perlmutter et al. 1999, ApJ, 517, 56

Inverse flux quantum periodicity of magnetoresistance oscillations in two-dimensional short-period surface superlattices

Transport properties of the two-dimensional electron gas (2DEG) are considered in the presence of a perpendicular magnetic field $B$ and of a {\it weak} two-dimensional (2D) periodic potential modulation in the 2DEG plane. The symmetry of the latter is rectangular or hexagonal. The well-known solution of the corresponding tight-binding equation shows that each Landau level splits into several subbands when a rational number of flux quanta $h/e$ pierces the unit cell and that the corresponding gaps are exponentially small. Assuming the latter are closed due to disorder gives analytical wave functions and simplifies considerably the evaluation of the magnetoresistivity tensor $\rho_{\mu\nu}$. The relative phase of the oscillations in $\rho_{xx}$ and $\rho_{yy}$ depends on the modulation periods involved. For a 2D modulation with a {\bf short} period $\leq 100$ nm, in addition to the Weiss oscillations the collisional contribution to the conductivity and consequently the tensor $\rho_{\mu\nu}$ show {\it prominent peaks when one flux quantum $h/e$ passes through an integral number of unit cells} in good agreement with recent experiments. For periods $300- 400$ nm long used in early experiments, these peaks occur at fields 10-25 times smaller than those of the Weiss oscillations and are not resolved

Non-Commutative Inflation

We show how a radiation dominated universe subject to space-time quantization may give rise to inflation as the radiation temperature exceeds the Planck temperature. We consider dispersion relations with a maximal momentum (i.e. a mimimum Compton wavelength, or quantum of space), noting that some of these lead to a trans-Planckian branch where energy increases with decreasing momenta. This feature translates into negative radiation pressure and, in well-defined circumstances, into an inflationary equation of state. We thus realize the inflationary scenario without the aid of an inflaton field. As the radiation cools down below the Planck temperature, inflation gracefully exits into a standard Big Bang universe, dispensing with a period of reheating. Thermal fluctuations in the radiation bath will in this case generate curvature fluctuations on cosmological scales whose amplitude and spectrum can be tuned to agree with observations.Comment: 4 pages, 3 figure

Scalar field exact solutions for non-flat FLRW cosmology: A technique from non-linear Schr\"odinger-type formulation

We report a method of solving for canonical scalar field exact solution in a non-flat FLRW universe with barotropic fluid using non-linear Schr\"{o}dinger (NLS)-type formulation in comparison to the method in the standard Friedmann framework. We consider phantom and non-phantom scalar field cases with exponential and power-law accelerating expansion. Analysis on effective equation of state to both cases of expansion is also performed. We speculate and comment on some advantage and disadvantage of using the NLS formulation in solving for the exact solution.Comment: 12 pages, GERG format, Reference added. accepted by Gen. Relativ. and Gra

Magnetotransport in Two-Dimensional Electron Systems with Spin-Orbit Interaction

We present magnetotransport calculations for homogeneous two-dimensional electron systems including the Rashba spin-orbit interaction, which mixes the spin-eigenstates and leads to a modified fan-chart with crossing Landau levels. The quantum mechanical Kubo formula is evaluated by taking into account spin-conserving scatterers in an extension of the self-consistent Born approximation that considers the spin degree of freedom. The calculated conductivity exhibits besides the well-known beating in the Shubnikov-de Haas (SdH) oscillations a modulation which is due to a suppression of scattering away from the crossing points of Landau levels and does not show up in the density of states. This modulation, surviving even at elevated temperatures when the SdH oscillations are damped out, could serve to identify spin-orbit coupling in magnetotransport experiments. Our magnetotransport calculations are extended also to lateral superlattices and predictions are made with respect to 1/B periodic oscillations in dependence on carrier density and strength of the spin-orbit coupling.Comment: 8 pages including 8 figures; submitted to PR

Could thermal fluctuations seed cosmic structure?

We examine the possibility that thermal, rather than quantum, fluctuations are responsible for seeding the structure of our universe. We find that while the thermalization condition leads to nearly Gaussian statistics, a Harrisson-Zeldovich spectrum for the primordial fluctuations can only be achieved in very special circumstances. These depend on whether the universe gets hotter or colder in time, while the modes are leaving the horizon. In the latter case we find a no-go theorem which can only be avoided if the fundamental degrees of freedom are not particle-like, such as in string gases near the Hagedorn phase transition. The former case is less forbidding, and we suggest two potentially successful ``warming universe'' scenarios. One makes use of the Phoenix universe, the other of ``phantom'' matter.Comment: minor corrections made, references added, matches the version accepted to PR

Particle-Like Description in Quintessential Cosmology

Assuming equation of state for quintessential matter: $p=w(z)\rho$, we analyse dynamical behaviour of the scale factor in FRW cosmologies. It is shown that its dynamics is formally equivalent to that of a classical particle under the action of 1D potential $V(a)$. It is shown that Hamiltonian method can be easily implemented to obtain a classification of all cosmological solutions in the phase space as well as in the configurational space. Examples taken from modern cosmology illustrate the effectiveness of the presented approach. Advantages of representing dynamics as a 1D Hamiltonian flow, in the analysis of acceleration and horizon problems, are presented. The inverse problem of reconstructing the Hamiltonian dynamics (i.e. potential function) from the luminosity distance function $d_{L}(z)$ for supernovae is also considered.Comment: 35 pages, 26 figures, RevTeX4, some applications of our treatment to investigation of quintessence models were adde

Is the Universe Inflating? Dark Energy and the Future of the Universe

We consider the fate of the observable universe in the light of the discovery of a dark energy component to the cosmic energy budget. We extend results for a cosmological constant to a general dark energy component and examine the constraints on phenomena that may prevent the eternal acceleration of our patch of the universe. We find that the period of accelerated cosmic expansion has not lasted long enough for observations to confirm that we are undergoing inflation; such an observation will be possible when the dark energy density has risen to between 90% and 95% of the critical. The best we can do is make cosmological observations in order to verify the continued presence of dark energy to some high redshift. Having done that, the only possibility that could spoil the conclusion that we are inflating would be the existence of a disturbance (the surface of a true vacuum bubble, for example) that is moving toward us with sufficiently high velocity, but is too far away to be currently observable. Such a disturbance would have to move toward us with speed greater than about 0.8c in order to spoil the late-time inflation of our patch of the universe and yet avoid being detectable.Comment: 7 pages, 7 figure