Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations

We study quasilinear elliptic equations of the type $$-\Delta_pu=\sigma \, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $\Delta_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $\sigma \ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian $(-\Delta )^{\alpha}$ for $0<\alpha<\frac{n}{2}$ are treated as well, including explicit estimates for radially symmetric $\sigma$. Our results are new even in the classical case $p=2$ and $\alpha=1$.Comment: 24 page

Spontaneous violation of mirror symmetry

A symmetry violation model is considered for a system that can spontaneously choose between identical states which differ from each other only in weak properties (R-L). Such mirror symmetry allows reproduction of observed qualitative properties of quark and lepton mixing matrices. The lepton mixing matrix evidences in this case in favor of the inverse mass spectrum and the Dirac nature of SM neutrino. Notwithstanding the Dirac properties of neutrino, an exchange of lepton numbers such as $e^{-}+\mu^{+}\rightarrow e^{+}+\mu^{-}$ is possible but with only leptons participating in the process.Comment: 10 pages, 3 figures, submitted to Yad.Fiz ( Phys.Atom.Nucl

Mechanism for enhancement of superconductivity in multi-band systems with odd parity hybridization

The study of multi-band superconductivity is relevant for a variety of systems, from ultra cold atoms with population imbalance to particle physics, and condensed matter. As a consequence, this problem has been widely investigated bringing to light many new and interesting phenomena. In this work we point out and explore a correspondence between a two-band metal with a $k$-dependent hybridization and a uniformly polarized fermionic system in the presence of spin-orbit coupling (SOC). We study the ground state phase diagram of the metal in the presence of an attractive interaction. We find remarkable superconducting properties whenever hybridization mixes orbitals of different parities in neighboring sites. We show that this mechanism enhances superconductivity and drives the crossover from weak to strong coupling in analogy with SOC in cold atoms. We obtain the quantum phase transitions between the normal and superfluid states, as the intensity of different parameters characterizing the metal are varied, including Lifshitz transitions, with no symmetry breaking, associated with the appearance of soft modes in the Fermi surface.Comment: 19 pages, 10 figure

Many-body spin interactions and the ground state of a dense Rydberg lattice gas

We study a one-dimensional atomic lattice gas in which Rydberg atoms are excited by a laser and whose external dynamics is frozen. We identify a parameter regime in which the Hamiltonian is well-approximated by a spin Hamiltonian with quasi-local many-body interactions which possesses an exact analytic ground state solution. This state is a superposition of all states of the system that are compatible with an interaction induced constraint weighted by a fugacity. We perform a detailed analysis of this state which exhibits a cross-over between a paramagnetic phase with short-ranged correlations and a crystal. This study also leads us to a class of spin models with many-body interactions that permit an analytic ground state solution

Nonlinear transverse magnetic moment in anisotropic superconductors

We consider the nonlinear transverse magnetic moment that arises in the Meissner state of superconductors with strongly anisotropic order parameter. We compute this magnetic moment as a function of applied field and geometry, assuming d-wave pairing, for realistic samples, finite in all three dimensions, of high temperature superconducting materials. Return currents, shape effects and the anisotropy of the penetration depth are all included. We numerically solve the nonlinear Maxwell-London equations for a finite system. The effect, which is a probe of the order parameter symmetry in the bulk of the sample, should be readily measurable if pairing is in a d-wave state. Failure to observe it would set a lower bound to the s-wave component.Comment: RevTex, 15 pages, six Postscript Figure