## 8 Conclusions  
The main results of this paper include the discovery of the existence of generalized adiabatic invariants and a generalized Born-Fock adiabatic theorem for a charged particle in a homogeneous magnetic field, when this field slowly passes through zero value. According to the original Born-Fock theorem, the initial discrete energy eigenstates maintain their forms during the adiabatic evolution of the Hamiltonian, so that parameters of the wave functions correspond to instantaneous values of coefficients of the Hamiltonian. This statement becomes invalid when the time-dependent Larmor frequency approaches zero value. We considered a scenario where the frequency slowly departs from zero, in such a way that the adiabatic condition (20) is reestablished. The evolved quantum state then becomes a _superposition_ of instantaneous energy eigenstates with different radial quantum numbers (but the same angular quantum numbers, due to the conservation of the canonical angular momentum). The generalized Born-Fock theorem states that the weight of each member of this superposition does not depend on time, as soon as the condition (20) is fulfilled. It is remarkable that all these weights depend on the single parameter \(|u_{-}|\) of the most general adiabatic solution (21) to the classical Equation (7) of the harmonic oscillator with a time-dependent frequency. Using examples of exact solutions, we believe (although we have no rigorous proof) that the value of \(|u_{-}|\) in the case of a single transition of frequency through zero is determined by the exponent \(k\) in the frequency behavior near zero: \(\omega(t)\sim(t-t_{*})^{k}\). In particular, \(|u_{-}|=1\) if \(k=1\). Regarding cases where there are multiple frequency passages through zero, the final coefficient \(|u_{-}|\) is highly sensitive to the additional parameter--the phase \(\Phi\), which is the integral of the absolute value \(|\omega(t)|\) between the first and last zero frequency instants. Consequently, the adiabatic behavior after many crossings through zero frequency can be quasi-chaotic. Under certain specific conditions, the mean energy can even return to the initial value after multiple frequency passage through zero, while it can be significantly amplified under other specific conditions. However, in all the cases, the adiabatic ratio \(\langle\hat{H}(t)\rangle/[\hbar|\omega(t)|]\) can only increase after many frequency passages through zero value. This ratio coincides (by absolute value) with the double mean value of the magnetic moment in the adiabatic regime. The word \"double\" means the quantum-mechanical averaging accompanied with the time averaging over rapid oscillations in time. Note that the concepts of adiabaticity and multiple passages through zero frequency are completely compatible. For example, the Larmor frequency of an electron in the magnetic field of the order of 1 T is of the order of \(10^{11}\) s\({}^{-1}\). In such a case, periodic variations of the magnetic field with the standard frequency 50 Hz are totally adiabatic. It would be interesting to observe what can happen with the statistical properties of the energy and magnetic moment in this realistic situation after a great number (of an order of \(10^{2}\) or higher) of frequency transitions through zero. However, this can be the subject of another study. There are several other challenging problems. In particular, in the case of very slow evolution, various _dissipation_ effects can change the evolution drastically. Another problem to address is gaining an understanding of the adiabatic evolution for different gauges of the vector potential. Some results presented in the paper by Ref. [36] indicate that the evolution can be quite different for the Landau gauge, because of another geometry of the induced electric field. However, the general linear gauge of the time-dependent magnetic field remains an unsolved problem.