The Vlasov equation is used to determine the dispersion relation for the eigenmodes of magnetized nuclear and neutral stellar matter, taking into account the anomalous magnetic moment of nucleons. The formalism is applied to the determination of the dynamical spinodal section, a quantity that gives a good estimation of the crust-core transition in neutron stars. We study the effect of strong magnetic fields, of the order of 10 15 -10 17 G, on the extension of the crust of magnetized neutron stars. The dynamical instability region of neutron-proton-electron (npe) matter at subsaturation densities is determined within a relativistic mean-field model. It is shown that a strong magnetic field has a large effect on the instability region, defining the crust-core transition as a succession of stable and unstable regions due to the opening of new Landau levels. The effect of the anomalous magnetic moment is non-negligible for fields larger than 10 15 G. The complexity of the crust at the transition to the core and the increase of the crust thickness may have direct impact on the properties of neutrons stars related with the crust.

We study the simultaneous effects of the symmetry energy and temperature on the crust-core transition of a magnetar. The dynamical and the thermodynamical spinodals are used to calculate the transition region within a relativistic mean-field approach for the equation of state. Quantizing magnetic fields with intensities in the range of 2 × 10 15 < B < 5 × 10 16 G are considered. Under these strong magnetic fields, the crust extension is very sensitive to the density dependence of the symmetry energy, and the properties that depend on the crust thickness could set a constraint on the equation of state. It is shown that the effect on the extension of the crust-core transition is washed out for temperatures above 10 9 K. However, for temperatures below that value, a noticeable effect exists that grows as the temperature decreases and which should be taken into account when the evolution of magnetars is studied.

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with highorder regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

In this work we compute bound solutions for particles and anti-particles of the Dirac equation for a pure tensor radial Coulomb potential plus a constant. We find that the binding depends on the sign the tensor constant potential, and allows only bound solutions for a certain sign and magnitude of the κ quantum number, which is related to the spin-orbit coupling in the Dirac equation. This relation is reversed for anti-particle solutions. On the other hand, the Coulomb tensor potential, although not biding, changes the range of κ values for which there are bound solutions.

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