Neutral Fermion with a Magnetic Moment in External Electromagnetic Fields

The interaction between a massive neutral fermion with a static (spin) magnetic dipole moment and an external electromagnetic field is described by the Dirac–Pauli equation. Exact solutions of this equation are obtained along with the corresponding energy spectrum for an axially symmetric external magnetic field and for some centrally symmetric electric fields. It is shown that the spin–orbital interaction of a neutral fermion with a magnetic moment determines both the characteristic properties of the quantum states and the fermion energy spectrum. It is found that (1) the discrete energy spectrum of a neutral fermion depends on the projection of the fermion spin on a certain quantization axis, (2) the ground energy level of a fermion in these electric fields as well as the energy levels of all bound states with a fixed value of the quantum number characterizing the projection of the fermion spin in the electric field E = er is degenerate and the degeneration order is countably infinite, and (3) the energy spectra of neutral fermions and antifermions with spin magnetic moments are symmetric in centrally symmetric fields. Bound states of a neutral fermion with a magnetic moment in an external electric field do exist even if the Dirac–Pauli equation does not explicitly contain the term with the fermion mass. In addition, in centrally symmetric electric fields, there exist a countably infinite set of pairs of isolated charge-conjugate zero-energy solutions of the Dirac–Pauli equation.     

Dispersion Relations for Threshold Reactions in an External Electromagnetic Field

Nonlinear threshold phenomena in quantum systems in a strong external electromagnetic field are investigated. A development of traditional quantum electrodynamics (QED) and a modified QED with new universal parameter, fundamental mass, are used. Analysis of the threshold phenomena in an external magnetic field is based on the modified Dirac equation in an external field. The analysis shows that the fundamental mass determines a bound on the external field strength. Dispersion relations are proved for some threshold reactions in rather general external electromagnetic fields.     

Asymptotic Properties of the Electromagnetic Field in the External Schwarzschild Spacetime

. We study the asymptotic behaviour of the solutions to the vacuum Maxwell equations in the external Schwarzschild spacetime. The results are based on the extensive use of geometric considerations and the introduction of generalized energy estimates. We obtain the asymptotic behaviour along the null outgoing directions and we prove also some partial results concerning the behaviour along the timelike curves. Our techniques can be also used to control the asymptotic behaviour of the various derivatives of the Maxwell field and to obtain the asymptotic behaviour of the Weyl tensor fields, solutions of the "spin 2" equations.     

Investigation of Convection Flows in an Electromagnetic Field

We investigate the solvability of an initial-boundary-value problem for a system of equations of hydrodynamics taking into account the heat transfer and displacement currents in the Maxwell equation system. Under certain conditions, we prove the global (in time) unique solvability of this problem in weighted functional spaces. Moreover, we prove that the problem can be considered as a regularly perturbed initial-boundary-value problem in which the electroconductivity of a solid medium plays the role of a small parameter.Dedicated to the Memory of Olga Aleksandrovna LADYZHENSKAYA__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 493–545, October–December, 2004.     

Green Equilibrium Measures and Representations of an External Field

We establish a representation for external fields involving Green potentials. This is the analogue of the representation of Rakhmanov and Buyarov involving logarithmic potentials. We also establish related results and present an example.     

A Time-dependent Interaction Problem Between an Electromagnetic Field and an Elastic Body

This paper considers the scattering of a time-dependent electromagnetic plane wave by a bounded elastic body.An appropriate decomposition of the coupling interface conditions is proposed according to the Voigt’s model between the electromagnetic and elastic medium.The original unbounded scattering problem is equivalently reduced into an initial-boundary value problem in a bounded domain by introducing an exact transparent boundary condition(TBC)on a sufficiently large sphere.Making use of the Lax-Milgram lemma,the abstract inversion theorem of Laplace transform and the energy method,we verify the well-posedness and stability for the reduced problem.Moreover,a priori estimates are established for the electromagnetic field and elastic displacement by taking special test functions directly in the time domain variational formulation.     

Potts Model on the Bethe Lattice with Nonmagnetic Impurities in An External Magnetic Field

We obtain a solution for the Potts model on the Bethe lattice in an external magnetic field with movable nonmagnetic impurities. Using the method of “pseudochaotic” impurity distribution (correlations in the positions of the impurity atoms for the neighboring sides vanish), we obtain a system of equations defining the first-order phase transition curve on the “temperature–external field” plane. We find the dependence of the endpoint of the phase transition line on the concentration of magnetic atoms.     

Support Theorems and an Injectivity Result for Integral Moments of a Symmetric m-Tensor Field

In this work, we show an injectivity result and support theorems for integral moments of a symmetric m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of symmetric m-tensor fields were first introduced by Sharafutdinov. First we generalize a Helgason type support theorem proven by Krishnan and Stefanov (Inverse Probl Imaging 3(3):453–464, 2009). We use this extended result along with the first integral moments of a symmetric m-tensor field to prove the aforementioned results.

     

Construction of Exact Solutions for Equilibrium Configurations of the Boundary of a Conducting Liquid Deformed By an External Electric Field

In a two-dimensional plane-symmetric formulation, we consider the problem of the equilibrium configurations of the free surface of a conducting capillary liquid placed in an external electric field. We find a one-parameter family of exact solutions of the problem according to which the fluid takes the shape of a blade. Such a configuration provides formally unlimited local field amplification: the field strength is maximum at the edge of the blade and drops to zero at the periphery. For a given potential difference between the liquid and the flat electrode located above it, we find threshold values of the electric field strength at the edge of the liquid blade, the radius of curvature of the edge, and the distance from the edge to the electrode limiting the region of existence of the solutions.     

On the Vacuum Polarization Density Caused by an External Field

We consider an external potential, – , due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density, as derived in physics literature, after a well-known renormalization procedure, depends decisively on the strength of . For small , more precisely as long as the lowest eigenvalue, e₁(), of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density vanishes. In other words the vacuum stays neutral. But as soon as e₁() dives into the lower continuum the vacuum gets spontaneously charged with charge 2e. Global charge conservation implies that two positrons were emitted out of the vacuum, this is, a large enough external potential can produce electron-positron pairs.We give a rigorous proof of that phenomenon.Communicated by Rafael D. Benguriasubmitted 18/12/03, accepted 08/04/04     

Determination of Permittivity from the Modulus of the Electric Strength of a High-Frequency Electromagnetic Field

Doklady Mathematics - For an unmagnetized nonconducting medium, a system of electrodynamic equations corresponding to time-periodic oscillations is considered. For this system, we study the problem...     

Decay of Quasi-Stationary States of Nonrelativistic Quantum Systems in an Intense Electromagnetic Field

We investigate the influence of intense electromagnetic fields on the formation and decay of quasi-stationary states of different quantum systems. Nonlinear equations that determine complex energy values are obtained from the expressions for the probability of the processes in an external electromagnetic field using the analytic continuation method. We obtain asymptotic expansions describing the action of weak and strong electromagnetic fields. If the distances that determine the formation of the processes in superstrong fields decrease significantly, this may lead to effects whose characteristic length scale corresponds to the formalism of the modified QED, i.e., QED with fundamental mass.     

Nonlinear Effects Induced by Vacuum Polarization by a Strong Electromagnetic Field in a Weak Static Electromagnetic Field

In the one-loop approximation of quantum electrodynamics, we study the Faraday rotation effect in the propagation of a weak linearly polarized wave through the field of a strong plane electromagnetic wave and also processes induced by a strong wave: the emission of a photon whose energy is an integer multiple of the strong-wave quantum and photon absorption with the emission of several photons identical (coherent) to the strong-wave quanta. The induced processes should occur in a vacuum in a superposition of external fields, namely, the field of a strong plane electromagnetic wave and a static spatially inhomogeneous electromagnetic field. We show that the induced photon emission in the field of a strong electromagnetic wave with circular polarization and in a static inhomogeneous field is equivalent to the merging of two strong-wave quanta with the production of one photon and that the induced photon absorption in the same combination of fields is equivalent to the decay (splitting) of a photon into two photons completely identical to the strong-wave quanta. All these effects are induced by the phenomenon of the vacuum polarization by a strong external time-dependent electromagnetic field. The probabilities of the induced processes are nontrivial nonlinear functions of the squared strength of the time-dependent field also depending on other parameters.     

Nonlinear Electromagnetic Delay of Electromagnetic Signals Propagating in the Magnetic Meridian Plane of Pulsars and Magnetars

We analyze the propagation of electromagnetic waves in the magnetic meridian planes of neutron stars with a strong magnetic field in the framework of the parameterized post-Maxwellian electrodynamics of the vacuum. The origin of these electromagnetic waves is the curvature emission of X-rays and gamma rays from high-energy electrons in the vicinity of the magnetic poles of neutron stars. We show that in the case of a slowly varying intensity of X-ray and gamma-ray emission, the delay of the slow normal mode of electromagnetic waves relative to the fast mode results in a shift of the time dependence of the intensity of the detected radiation with one polarization relative to that of the radiation with the orthogonal polarization. In the case of single X-ray or gamma-ray pulses, the delay effect results in the polarization of the detected pulse varying during the pulse length, the leading edge of all pulses being polarized normally to the magnetic equator plane of the neutron star. We note that the modern level of the experimental technique, in principle, allows observing the manifestations of the delay effect for signals of different polarizations.     

Multipolarons in a Constant Magnetic Field

The binding of a system of N polarons subject to a constant magnetic field of strength B is investigated within the Pekar–Tomasevich approximation. In this approximation, the energy of N polarons is described in terms of a non-quadratic functional with a quartic term that accounts for the electron–electron self-interaction mediated by phonons. The size of a coupling constant, denoted by α, in front of the quartic term is determined by the electronic properties of the crystal under consideration, but in any case it is constrained by 0 < α < 1. For all values of N and B, we find an interval α _N,B <  α <  1 where the N polarons bind in a single cluster described by a minimizer of the Pekar–Tomasevich functional. This minimizer is exponentially localized in the N-particle configuration space ${\mathbb{R}^{3N}}$ .     

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

We consider a nematic liquid crystal occupying the exterior region in \({\mathbb {R}}^3\) outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to an external field, modeled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough, we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.     

外界场作用下的波前曲率关系和波形变化

研究外界场作用下,激发介质的波前曲率关系和波形变化·　理论分析波前曲率关系,得到外界场作用下,波前的法向速度与波前平均曲率、平面波速和外界场之间的线性关系·　数值分析外界场作用下,激发介质的Bar_Eiswirth模型,得到外界场作用下激发介质的波形图案·　这里的理论分析、数值结果与BZ反应的实验结果完全一致,从而解释了外界场作用下的BZ现象,且由数值结果可知激发介质包含丰富的波形图案·     

Quantum Field Theory in Static External Potentials and Hadamard States

We prove that the ground state for the Dirac equation on Minkowski space in static, smooth external potentials satisfies the Hadamard condition. We show that it follows from a condition on the support of the Fourier transform of the corresponding positive frequency solution. Using a Klein space formalism, we establish an analogous result in the Klein–Gordon case for a wide class of smooth potentials. Finally, we investigate overcritical potentials, i.e. which admit no ground states. It turns out, that numerous Hadamard states can be constructed by mimicking the construction of ground states, but this leads to a naturally distinguished one only under more restrictive assumptions on the potentials.     

New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

In this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval [−1, 1], in the presence of an external fieldV. In particular, we show that ifVisC², then the equilibrium measure is absolutely continuous with respect to Lebesgue measure, with a density which is Hölder- on (−1, 1), and with at worst a square root singularity at ±1. Moreover, ifVis real analytic then the support of the equilibrium measure consists of a finite number of intervals. In the cases whereV=tx^m,m=1, 2, 3, or 4, the equilibrium measure is computed explicitly for allt∈R. For these cases the support of the equilibrium measure consists of 1, 2, or 3 intervals, depending ontandm. We also present detailed results for the general monomial caseV=tx^m, for allm∈N. The regularity results for the equilibrium measure are obtained by careful analysis of the Fekete points associated to the weighte^nV(x) dx. The results on the support of the equilibrium measure are obtained using two different approaches: (i) an explicit formula of the kind derived by physicists for mean-field theory calculations; (ii) detailed perturbation theoretic results of the kind that are needed to analyze the zero dispersion limit of the Korteweg–de Vries equation in Lax–Levermore theory. The implications of the above results for a variety of related problems in approximation theory and the theory of orthogonal polynomials are also discussed.