Exact solutions of Dirac equation in two spatial dimensions in the Coulomb field are obtained. Equation which determines the so-called critical charge of the Coulomb field is derived and solved for a simple model.

The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is discussed. This is connected to the problem of the two-dimensional hydrogen-like atom in the presence of external magnetic field. For weak magnetic fields, the approximate energy values are obtained by semiclassical method. In the case with strong magnetic fields, we present the exact recursion relations that determine the coefficients of the series expansion of wave functions, the possible energies and the magnetic fields. It is found that analytic solutions are possible for a denumerably infinite set of magnetic field strengths. This system thus furnishes an example of the so-called quasi-exactly solvable models. A distinctive feature in the Dirac case is that, depending on the strength of the Coulomb field, not all total angular momentum quantum number allow exact solutions with wavefunctions in reasonable polynomial forms. Solutions in the nonrelativistic limit with both attractive and repulsive Coulomb fields are briefly discussed by means of the method of factorization.

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A model of a degenerate neutron gas in chemical equilibrium with a background of degenerate electrons and protons in a constant uniform ultrastrong magnetic field is applied to describe the state of matter in the cores of strongly magnetized neutron stars. Expressions for the thermodynamic quantities are obtained including the anomalous magnetic moments of the fermions. It is shown that ͑1͒ the inclusion of the anomalous magnetic moments of charged fermions leads to nonperiodic magnetic oscillations of their thermodynamic quantities in strong magnetic fields, ͑2͒ the total stress energy tensor relevant for neutron star structure must include contributions from both the magnetized matter and the magnetic field and as a result the total pressure produced is anisotropic, and ͑3͒ complete spin polarization of neutrons occurring in superstrong magnetic fields must lead to an increase in the degeneracy pressure compared with the zero field case at the same neutron densities. It is hoped that the results obtained will have applications for the structure in neutron stars with ultrastrong frozen-in magnetic fields.Study of a relativistic electron gas in a strong magnetic field was stimulated by the discovery of magnetic fields of the order of Bу10 13 G at the neutron star surface ͓1-3͔. Such a magnetic field ''frozen in'' a neutron star may become much stronger in its central domain. Gravitational collapse of macroscopic magnetized bodies, composed of neutrons in a strong magnetic field B, may lead to extremely magnetized neutron stars, or magnetars ͓4,5͔. Their magnetic fields at the star surface are estimated to be of the order of Bу10 15 G ͓6,7͔ and the magnetic induction at the star core may go up to 10 18 G ͓8͔. The magnetic induction B needed to affect neutron star structure directly was estimated in ͓9͔ to be Bϳ2ϫ10 18 (M /1.4M ᭪ )(R/10 km) Ϫ2 G, where M and R are, respectively, the neutron star mass and radius.Recently, many works have been concerned with the effect of strong magnetic fields on elementary processes occurring at the star core. The behavior of a relativistic nucleon and electron gas in a constant strong magnetic field was studied in ͓10͔. The equation of state and the magnetization of relativistic fermions in strong uniform magnetic fields including the anomalous magnetic moments of the fermions were partly discussed in ͓11-14͔. The effect of strong magnetic fields on dense neutron-star matter was studied in ͓15,16͔ in the mean field approximation. However, contributions from the magnetic field energy density and pressure were not included ͓15,16͔ while the anisotropic effect of uniform magnetic fields either was not given enough attention ͓11,12͔ or was considered incorrectly ͓13,14͔.As is known ͓17,18͔ the chemical equilibrium in a degenerate gas of neutrons ͑n͒, protons ͑p͒, and electrons ͑e͒ must take into account the direct URCA processes pϩe→nϩ, n→pϩeϩ ͑ and denote the neutrino and antineutrino, respectively͒ in which the total number of baryons n b ϭn p ϩn n is conserved and the electroneutra...

Bound states of massive fermions in AharonovBohm (AB)-like fields have analytically been studied. The Hamiltonians with the (AB)-like potentials are essentially singular and therefore require specification of a one-parameter self-adjoint extension. We construct selfadjoint Dirac Hamiltonians with the AB potential in 2+1 dimensions that are specified by boundary conditions at the origin. It is of interest that for some range of the extension parameter the AB potential can bind relativistic charged massive fermions. The bound-state energy is determined by the AB magnetic flux and depends upon the fermion spin and extension parameter; it is a periodical function of the magnetic flux. We also construct self-adjoint Hamiltonians for the so-called Aharonov-Casher (AC) problem, show that nonrelativistic neutral massive fermions can be bound by the (AC) background, determine the range of the extension parameter in which fermion bound states exist, and find their energies as well as wave functions.