A quantization method based on the use of lowering and raising operators is developed and applied to describing states of Fermi particles that move under extreme external conditions (strong magnetic field and dense matter). The efficiency of this method is demonstrated by applying it to examples of finding exact solutions of quantum equations that describe the motion of charged particles in a magnetic field and dense matter. For the first time, the problem of charged-fermion motion in matter and an external magnetic field is formulated and solved with allowance for the anomalous magnetic moment of the particle. Exact solutions for the wave functions and energy spectrum of the respective modified Dirac equation are obtained. The application of these results to describing fermions and neutrinos is of special interest for astrophysical applications.

We briefly review the application of the method of exact solutions of quantum equations for the description of charged particle motion in external fields. We develop a quantization technique by using the lowering and raising operators to describe the states of fermions moving in a magnetic field and a dense medium. For the first time we have posed and solved the problem of a charged fermion moving in a medium and an external magnetic field by taking into account the anomalous magnetic moment of the particle: we have obtained exact solutions for the wave functions and energy spectrum of the corresponding modified Dirac equation. We discuss a quasi-classic interpretation of the solutions obtained and, in particular, find a correction for the synchrotron radiation intensity dependent on medium density.

We consider a problem of electron motion in different media and magnetic fields. It is shown that in the case of an immovable medium and constant homogenous magnetic field the electron energies are quantized. We also discuss the general problem of eigenvectors and eigenvalues of a given class of Hamiltonians. We examine obtained exact solutions for the particular case of the electron motion in a rotating neutron star which account for matter and magnetic field effects. We argue that all of these considerations can be useful for astrophysical applications, in particular for the description of electrons' and neutrinos motion in different environments.

A special vector functional space is defined for a weak formulation of the diffraction problem in a cone. For this space, a number of embedding theorems are proved. It is also shown that the diffraction problem is reduced to the Fredholm equation.