| Abstract: | The paper presents a brief review of the scientific work performed by the authors in the field of quantum mechanics and atomic, laser, and mathematical physics. The following problems are considered: the semiclassical theory of tunneling and multiphoton ionization of atoms and ions in a strong electromagnetic field; generalization of the Keldysh ionization theory to the relativistic case; calculation of the Coulomb corrections to the ionization rate of atoms for arbitrary values of the adiabaticity parameter γ: from γ ≪ 1 (the adiabatic region) to γ ≫ 1, when the laser field changes its direction and magnitude many times during the time of flight of the electron through the barrier; the Lorentz ionization of atoms moving in a constant magnetic field; the WKB approximation and the imaginary time method for describing electron tunneling through a time-varying barrier; the Stark effect in a strong field; the energy spectrum of a hydrogen atom in a strong and superstrong magnetic field; quantization with account of the barrier transparency; creation of electron-positron pairs from vacuum in a constant electric or intense pulsed (laser) field and the dependence of the number of pairs on the intensity and frequency of the laser field; the Feynman method of disentanglement of noncommuting operators and its applications: transitions between atomic states in an alternating magnetic field (the Majorana problem); a quantum oscillator with time-dependent frequency; and a singular oscillator. The mathematical problems of quantum mechanics are considered: the fall of a particle to the center; modification of the Bohr-Sommerfeld quantization condition for potentials with a barrier and the Kramers matching conditions; divergence of perturbation series and their summation; eigenvalues of the Casimir operators for irreducible representations of Lie groups, including the SU(2), SU(3), and SU(6) groups, which are widely used in physics. |
