Discrete quantum modes of the Dirac field in AdSd+1AdS_{d+1} backgrounds

It is shown that the free Dirac equation in spherically symmetric static backgrounds of any dimensions can be put in a simple form using a special version of Cartesian gauge in Cartesian coordinates. This is manifestly covariant under the transformations of the isometry group so that the generalized spherical coordinates can be separated in terms of angular spinors like in the flat case, obtaining a pair of radial equations. In this approach the equation of the free field Dirac in $AdS_{d+1}$ backgrounds is analytically solved obtaining the formula of the energy levels and the corresponding normalized eigenspinors.Comment: 18 pages, Latex. Submitted to Phys.Rev.

Approximative analytical solutions of the Dirac equation in Schwarzschild spacetime

Approximative analytic solutions of the Dirac equation in the geometry of Schwarzschild black holes are derived obtaining information about the discrete energy levels and the asymptotic behavior of the energy eigenspinors.Comment: 8 page

On the convergence of second order spectra and multiplicity

Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper

Maximal extension of the Schwarzschild spacetime inspired by noncommutative geometry

We derive a transformation of the noncommutative geometry inspired Schwarzschild solution into new coordinates such that the apparent unphysical singularities of the metric are removed. Moreover, we give the maximal singularity-free atlas for the manifold with the metric under consideration. This atlas reveals many new features e.g. it turns out to describe an infinite lattice of asymptotically flat universes connected by black hole tunnels.Comment: 17 pages LaTex, 2 figure

Vacuum energy between a sphere and a plane at finite temperature

We consider the Casimir effect for a sphere in front of a plane at finite temperature for scalar and electromagnetic fields and calculate the limiting cases. For small separation we compare the exact results with the corresponding ones obtained in proximity force approximation. For the scalar field with Dirichlet boundary conditions, the low temperature correction is of order $T^2$ like for parallel planes. For the electromagnetic field it is of order $T^4$. For high temperature we observe the usual picture that the leading order is given by the zeroth Matsubara frequency. The non-zero frequencies are exponentially suppressed except for the case of close separation.Comment: 14 pages, 3 figures, revised version with several improvement

Birefringent Gravitational Waves and the Consistency Check of Inflation

In this work we show that the gravitational Chern-Simons term, aside from being a key ingredient in inflationary baryogenesis, modifies super-horizon gravitational waves produced during inflation. We compute the super-Hubble gravitational power spectrum in the slow-roll approximation and show that its overall amplitude is modified while its spectral index remains unchanged (at leading order in the slow-roll parameters). Then, we calculate the correction to the tensor to scalar ratio, T/S. We find a correction of T/S which is dependent on $\cal{N}$ (more precisely quadratic in ${\cal N}$), the parameter characterizing the amplitude of the Chern-Simons terms. In a stringy embedding of the leptogenesis mechanism, $\cal{N}$ is the ratio between the Planck scale and the fundamental string scale. Thus, in principle, we provide a direct probe of leptogenesis due to stringy dynamics in the Cosmic Microwave Background (CMB). However, we demonstrate that the corresponding correction of T/S is in fact very small and not observable in the regime where our calculations are valid. To obtain a sizable effect, we argue that a non-linear calculation is necessary.Comment: 9 pages, 1 figure, RevTe

Position and Momentum Uncertainties of the Normal and Inverted Harmonic Oscillators under the Minimal Length Uncertainty Relation

We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by [x,p]=i\hbar(1+\beta p^2). This deformed commutation relation leads to the minimal length uncertainty relation \Delta x > (\hbar/2)(1/\Delta p +\beta\Delta p), which implies that \Delta x ~ 1/\Delta p at small \Delta p while \Delta x ~ \Delta p at large \Delta p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0), derived in Ref. [1], only populate the \Delta x ~ 1/\Delta p branch. The other branch, \Delta x ~ \Delta p, is found to be populated by the energy eigenstates of the `inverted' harmonic oscillator (m<0). The Hilbert space in the 'inverted' case admits an infinite ladder of positive energy eigenstates provided that \Delta x_{min} = \hbar\sqrt{\beta} > \sqrt{2} [\hbar^2/k|m|]^{1/4}. Correspondence with the classical limit is also discussed.Comment: 16 pages, 31 eps figure

Quasi-Ferromagnet Spintronics in Graphene Nanodisk-Lead System

A zigzag graphene nanodisk can be interpreted as a quantum dot with an internal degree of freedom. It is well described by the infinite-range Heisenberg model. We have investigated its thermodynamical properties. There exists a quasi-phase transition between the quasi-ferromagnet and quasi-paramagnet states, as signaled by a sharp peak in the specific heat and in the susceptability. We have also analyzed how thermodynamical properties are affected when two leads are attached to the nanodisk. It is shown that lead effects are described by the many-spin Kondo Hamiltonian. There appears a new peak in the specific heat, and the multiplicity of the ground state becomes just one half of the system without leads. Another lead effect is to enhance the ferromagnetic order. Being a ferromagnet, a nanodisk can be used as a spin filter. Furthermore, since the relaxation time is finite, it is possible to control the spin of the nanodisk by an external spin current. We then propose a rich variety of spintronic devices made of nanodisks and leads, such as spin memory, spin amplifier, spin valve, spin-field-effect transistor, spin diode and spin logic gates such as spin-XNOR gate and spin-XOR gate. Graphene nanodisks could well be basic components of future nanoelectronic and spintronic devices.Comment: 12 pages, 13 figures, invited paper to "focus on graphene

Theory of 4e versus 2e supercurrent in frustrated Josepshon-junction rhombi chain

We consider a chain of Josepshon-junction rhombi (proposed originally in \cite{Doucot}) in quantum regime, and in the realistic case when charging effects are determined by junction capacitances. In the maximally frustrated case when magnetic flux through each rhombi $\Phi_r$ is equal to one half of superconductive flux quantum $\Phi_0$, Josepshon current is due to correlated transport of {\em pairs of Cooper pairs}, i.e. charge is quantized in units of $4e$. Sufficiently strong deviation $\delta\Phi \equiv |\Phi_r-\Phi_0/2| > \delta\Phi^c$ from the maximally frustrated point brings the system back to usual $2e$-quantized supercurrent. We present detailed analysis of Josepshon current in the fluctuation-dominated regime (sufficiently long chains) as function of the chain length, $E_J/E_C$ ratio and flux deviation $\delta\Phi$. We provide estimates for the set of parameters optimized for the observation of $4e$-supercurrent.Comment: 23 pages, 9 figure

Approximate solutions of the Dirac equation for the Rosen-Morse potential including the spin-orbit centrifugal term

We give the approximate analytic solutions of the Dirac equations for the Rosen-Morse potential including the spin-orbit centrifugal term. In the framework of the spin and pseudospin symmetry concept, we obtain the analytic bound state energy spectra and corresponding two-component upper- and lower-spinors of the two Dirac particles, in closed form, by means of the Nikiforov-Uvarov method. The special cases of the s-wave kappa=1,-1 (l=l bar=0) Rosen-Morse potential, the Eckart-type potential, the PT-symmetric Rosen-Morse potential and non-relativistic limits are briefly studied.Comment: 23 page