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.     

Pauli isotonic oscillatorwith an anomalous magnetic moment in the presence of the Aharonov–Bohm effect: Laplace transform approach

A strong magnetic field significantly affects the intrinsic magnetic moment of fermions. In quantum electrodynamics, it was shown that the anomalous magnetic moment of an electron arises kinematically, while it results from a dynamical interaction with an external magnetic field for hadrons (proton). Taking the anomalous magnetic moment of a fermion into account, we find an exact expression for the boundstate energy and the corresponding eigenfunctions of a two-dimensional nonrelativistic spin-1/2 harmonic oscillator with a centripetal barrier (known as the isotonic oscillator) including an Aharonov–Bohm term in the presence of a strong magnetic field. We use the Laplace transform method in the calculations. We find that the singular solution contributes to the phase of the wave function at the origin and the phase depends on the spin and magnetic flux.     

Neutral Fermion Having Electric and Magnetic Moments in an External Electromagnetic Field

We show that in 2+1 dimensions, the Dirac equation for a neutral fermion possessing electric and magnetic dipole moments in an external electromagnetic field reduces to the Dirac equation for a charged fermion in a external field characterized by a certain 3-pseudo-vector potential. The effective charge of the neutral fermion is determined by its dipole moments. The effects of coupling electric and magnetic moments of the neutral fermion to the external electromagnetic field seem to be inseparable in physical experiments of any type. We find an exact solution of the Dirac equation for a massive neutral fermion with electric and magnetic dipole moments in a external plane-wave electromagnetic field. We derive expressions for the fermionic vacuum current induced by neutral fermions in the presence of external electromagnetic fields.     

Fermion bound states in the Aharonov-Bohm field in 2+1 dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in 2+1 dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions describe the behavior of relativistic fermions in the field of the cosmic string in 3+1 dimensions, our results can presumably be used to describe fermions in the cosmic string field.     

Scattering by magnetic nanocylinders and the method of distorted waves in noncollinear fields

In the perturbation theory framework, we compute the cross section of scattering by a magnetic nanocylinder and a helicoid arbitrarily oriented in an external magnetic field. We are the first to obtain the matrix Green’s function for two media with an interface and noncollinear magnetic fields on the two sides of the interface. We show how to compute scattering by magnetic inclusions in one of the media.     

Inverse Problems for the Pauli Hamiltonian in Two Dimensions

We prove in two dimensions that the set of Cauchy data for the Pauli Hamiltonian measured on the boundary of a bounded open subset with smooth enough boundary determines uniquely the magnetic field and the electrical potential provided that the electrical potential is small in an appropriate topology. This result has the immediate consequence, in the case that the magnetic potential and electrical potential have compact support, that we can determine uniquely the magnetic field and the electrical potential by measuring the scattering amplitude at a fixed energy provided that the electrical potential is small in an appropriate topology.     

Microscopic Quantum Hydrodynamics of Systems of Fermions: Part I

The fundamental equations of the microscopic quantum hydrodynamics of fermions in an external electromagnetic field (i.e., the particle balance equation, the momentum balance equation, the energy balance equation, and the magnetic moment balance equation) are derived using the Schrödinger equation. The form of the spin–spin interaction Hamiltonian is specified. To close the system of the balance equations for a multiparticle fermion system, the effective one-particle Schrödinger equation must be introduced.     

Equilibrium of a Degenerate Gas of Nucleons and Electrons in a Strong Magnetic Field

A model of a degenerate ideal gas of nucleons and electrons in a superstrong magnetic field is used to describe the state of matter in the central region of a strongly magnetized neutron star. The influence of a constant uniform superstrong magnetic field on the equilibrium conditions and the equation of state for the degenerate gas of neutrons, protons, and electrons is investigated in the framework of this model. The contribution determined by the interaction of the anomalous magnetic moments of the fermions with the magnetic field is taken into account. The influence of the superstrong magnetic field on the process of gravitational collapse of a magnetized neutron star is discussed under the assumption that the central region of the star consists mostly of degenerate neutrons. We show that if the densities of electrons, protons, and neutrons are relatively low depending on the field strength, the fermion gases in a superstrong uniform magnetic field become totally polarized with respect to the spin. We discuss the possibility of spontaneous magnetization occurring in a system of degenerate neutrons where the exchange interaction effects are taken into account.     

The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x₃‐axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x₃‐axis, we consider the reduced Hamiltonian associated with the total momentum along the x₃‐axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x₃‐axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is nonnegative. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L²(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.     

The resistive coated sphere in the presence of a point generated wave field

We consider the low‐frequency scattering problem of a point source generated incident field by a small penetrable sphere. The sphere, which is also lossy, contains in its interior a co‐ecentric spherical core on the boundary of which an impedance boundary condition is satisfied. An appropriate modification of the incident wave field allows for the reduction of the solution to the corresponding scattering problem of plane wave incidence, by moving the point source to infinity. For the near field, we obtain the low‐frequency coefficients of the zeroth and the first order. This was done with the help of the corresponding solution for the hard core problem and an appropriate use of linearity with respect to the Robin parameter. In the far field, we derive the leading non‐vanishing terms for the normalized scattering amplitude and the scattering cross‐section, which are both of the second order, as well as for the absorption cross‐section, which is of the zeroth order. The special cases of a lossy or a lossless penetrable sphere, of a resistive sphere, and of a hard sphere are recovered by an appropriate choice of the physical or the geometrical parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

Electromagnetic scattering theorems for interior dipole excitation of a layered obstacle

A layered scatterer is excited by a time‐harmonic spherical electromagnetic wave, generated by a dipole located either in the interior or in the exterior of the scatterer. The scatterer's core may be perfect conducting, impedance or dielectric. This paper initiates the investigation of scattering theorems corresponding to the excitation of a layered scatterer by a dipole in its interior. We establish reciprocity and general scattering theorems relating the total electric fields with the corresponding far‐field patterns. The optical theorem, relating the scattering cross‐section with the electric field in the layer containing the dipole, is recovered as a corollary of the general scattering theorem. Furthermore, for a scatterer excited by a spherical and a plane wave, mixed scattering theorems are derived. Besides, all the derived theorems recover known results, concerning the excitation of a homogeneous scatterer by an exterior dipole. We also analyze numerical implementations of the optical theorem in certain concrete scattering applications. Copyright © 2007 John Wiley & Sons, Ltd.     

Scattering of electromagnetic plane waves by a spheroid uniformly moving in free space

We investigate the scattering process, generated by a plane electromagnetic field that is incident upon a moving perfectly conducting spheroid. An accurate treatment of the electromagnetic waves interaction with scatterers in uniform motion is based on the special relativity principle. In the object's frame the incident wave is assumed to have a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low‐frequency approximation method is applicable to the scattering problem. For the near electromagnetic field we obtain the zeroth‐order low‐frequency coefficients, while in the far field we calculate the leading terms for the scattering amplitude and scattering cross‐section. Finally, using the inverse Lorentz transform, we obtain the same approximations in the observer's frame. Copyright © 2006 John Wiley & Sons, Ltd.     

Electrohydrodynamic flow in a retarding electric field with allowance for charged particle inertia : PMM vol. 43, no. 6, 1979, pp. 1029–1039

The one-dimensional electrohydrodynamic flow is considered in a retarding electric field when, owing to the inertia of charged particles, it is necessary to use the complete equation of momenta for the charged component. It is shown that in spite of the negligibly small relative volume occupies by particles in the initial cross section of the stream, there is a section in which particles move at low but finite velocity with a finite relative volume of the charged component, when the particles are retarded by an external homogeneous electric field (without taking into account the induced field). Interaction of the charged and neutral components, which may be absent in the initial cross section, is always substantial. The pressure drop required for such flow is determined. The flow is investigated with allowance for induced electric fields that diminish the effect of concentration increase of charged particles.     

Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

Indefinite Boundary Eigenvalue Problems in a Pontrjagin Space Setting

We study eigenvalue problems Fy = λ Gy consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric λ-linear boundary conditions. The problems we are interested in are non-definite: neither left-nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a hermitian form associated with the operator F has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontrjagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher order are given.     

Properties of the heavy-fermion spectrum in the canted phase of antiferromagnetic intermetallides

We study the effect of the magnetic-field-induced canting of magnetic sublattices on the energy structure of heavy fermion quasiparticles in intermetallides with antiferromagnetic-type ordering. We work in the framework of an effective Hamiltonian of the periodic Anderson model in the regime of strong electron correlations. With the virtual transfers into high-energy double states taken into account, this Hamiltonian involves exchange interactions between spin moments of the f ions and the s-f-exchange coupling between the two subsystems. For a noncollinear problem geometry, we introduce a unitary transformation that allows reducing an eighth-order equation for the spectrum of heavy fermions to two fourth-order equations. We show that the quasimomentum dependence of the heavy-fermion energy changes qualitatively under the transition from the antiferromagnetic phase to a phase with a considerable canting angle emerging in an external magnetic field.     

Eigenvalues behaviours for self-adjoint Pauli operators with unsigned perturbations and admissible magnetic fields

We investigate the discrete spectrum behaviour for the 2d Pauli operator with nonconstant magnetic field, perturbed by a sign-indefinite self-adjoint electric potential that decays polynomially at infinity. A localisation of the eigenvalues and new asymptotics are established.     

Quantum Macroscopic Effects in a Degenerate Strongly Magnetized Nucleon Gas

A model of a degenerate gas consisting of neutrons that are in chemical equilibrium with degenerate protons and electrons in a stationary and homogeneous superstrong magnetic field is used to describe the state of the matter in central regions of strongly magnetized neutron stars. Expressions for thermodynamic quantities (such as energy density, particle density, pressure, and magnetization) characterizing a degenerate gas of neutrons, protons, and electrons are obtained. In these expressions, the contributions determined by the interaction between anomalous magnetic moments of fermions and the magnetic field are taken into account. Macroscopic effects that may occur in strongly magnetized neutron stars are discussed. We show that all thermodynamic quantities characterizing electrically charged fermions in a strong magnetic field are subject to nonperiodic oscillations caused by the interaction of the anomalous magnetic moments of protons and electrons with the magnetic field. We also show that if the nucleon density and the electron density exceed threshold values that are relatively small and depend on the magnetic field strength, all fermions are fully polarized with respect to the spin. The full spin polarization effect in neutrons is caused by the interaction between the anomalous magnetic moment and the magnetic field. The obtained results may prove useful in understanding processes that occur in the nucleus of a neutron star with a magnetic field frozen into the star.     

On the Spectral Properties of the Perturbed Landau Hamiltonian

The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.