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In this paper we discuss in detail the quantization of Landau energy levelsof a strongly magnetized and completely degenerate relativistic electron gas inneutron stars. In particular, we focus on the Fermi energy dependence of themagnetic field for a relativistic electron gas in the superstrong magneticfield of magnetars. We would like to point out that some of the resultsconcerning the microscopic number density of states of a strongly magnetizedelectron gas given by well-known statistical physics text books are incorrect.The repeated use of these results in the last five decades probably seriouslyaffects the physics of neutron stars and magnetars. The quantization of Landauenergy levels is accurately delineated in terms of the Dirac (delta) function.Relatively simple calculation shows that the Fermi energy of relativisticelectron gas in magnetars with superstrong magnetic fields( B > Bcr , here Bcris the Landau critical magnetic field) increases with magnetic fieldstrength asB**(1/4). On the basis of this simple but importantnew result we are leadingnaturally to propose a new mechanism for the production of high X-rayluminosity from magnetars.

The Physics of Magnetars II - The Electron Fermi Energy of and the Origin of High X-ray Luminosity of Magnetars