Eigenvalues of the Operator Describing Magnetohydrodynamic Problems in Outer Parts of Galaxies

The magnetic field generation studies in astronomy lead to a number of interesting problems in mathematical physics. In the dynamo theory, the problem is reduced to a system of parabolic equations for the field components. Assuming that the field grows exponentially, we obtain an eigenvalue problem for the corresponding elliptic operator. The possibility of the field generation and behaviour of the system is characterized by the spectra of the operator. If all eigenvalues lie in the left half of the complex plane, the perturbations will decay. On the other hand, if some of the eigenvalues have positive real parts, the large-scale structures of the field can be generated. From the astrophysical point of view, galactic magnetic fields are very important to study. One of the main problems is connected with the peripheral regions, where the properties of the medium complicate the operator structure. We can use the perturbation theory to find the eigenvalues. However, the problem can be solved analytically by considering some specific approximations. We can find the spectra using numerical approaches in the case of the conditions that are close to the real ones. In this paper, we solve eigenvalue problems for different operators which are connected with magnetohydrodynamic processes in outer parts of galaxies.