## 3 Examples of Adiabatic Coefficients  
Exact values of the adiabatic coefficients are determined by the behavior of function \(\omega(t)\) near zero value. It was proposed in paper [23], that absolute values of these coefficients can be expressed in terms of the exponent \(n\) in the behavior \(\omega^{2}(t)\sim|t-t_{*}|^{n}\) when \(\omega(t)\) passes through zero. This hypothesis was based on the analysis of exact solutions to Equation (7) for the time-dependent frequency \(\omega^{2}(t)=\omega_{0}^{2}|t/\tau|^{n}\), when this equation can be reduced to the Bessel equation. In that paper, devoted to the adiabatic dynamics of a quantum harmonic oscillator, the exponent \(n\) could assume arbitrary non-negative values, as soon as function \(\omega^{2}(t)\) had physical meaning. However, it is the function \(\omega(t)\) that has physical meaning in the problems involving magnetic fields. For this reason, supposing that function \(\omega(t)\) can smoothly change its sign, we consider here the special case of solutions obtained in [23], when \(n\) is an _even_ number, i.e.,  
\[\omega(t)=\omega_{0}(-t/\tau)^{k},\qquad-\tau\leq t\leq\tau,\qquad k=0,1,2,\dots \tag{27}\]  
Then, in the adiabatic limit \(\omega_{0}\tau\gg 1\), the following formulas were derived in [23]:  
\[u_{+}^{-1}=\sin\!\bigg{[}\frac{\pi}{2(k+1)}\bigg{]},\qquad u_{-}=i\cot\!\bigg{[} \frac{\pi}{2(k+1)}\bigg{]}. \tag{28}\]  
It is probable that the most natural situation takes place for \(k=1\), i.e., the linear time dependence of a slow transition of magnetic field through zero value. In this case, we have \(|u_{-}|=1\) and \(|u_{+}|^{2}=2\). Another interesting situation corresponds to \(k=2\), when \(|u_{-}|^{2}=3\). In this case, the magnetic field slowly diminishes to zero, but instead of crossing through zero, it slowly returns to the initial value. For \(k=3\) (a \"cubical\" crossing through zero frequency), we have \(|u_{-}|^{2}=(\sqrt{2}+1)^{2}\).  
Another explicit exact solution to Equation (7) was found in paper [34] for the time-dependent Larmor frequency, which changes continuously from the initial value \(\omega_{i}\) to the final value \(\omega_{f}\) as  
\[\omega(t)=\frac{\omega_{f}\exp(\kappa t)+\omega_{i}}{\exp(\kappa t)+1},\quad- \infty<t<\infty,\quad\kappa>0. \tag{29}\]  
Here, the positive parameter \(\kappa\) characterizes the speed of frequency evolution. The following formula was obtained in the adiabatic regime, when \(|\omega_{i}-\omega_{f}|\gg\kappa/2\):  
\[|u_{-}|^{2}\approx\exp[2\pi(|\tilde{\omega}_{i}-\tilde{\omega}_{f}|-\tilde{ \omega}_{i}-|\tilde{\omega}_{f}|)],\qquad\tilde{\omega}_{j,k}\equiv\omega_{ i,k}/\kappa. \tag{30}\]  
If \(\omega_{i}>\omega_{f}>0\), Equation (30) yields \(|u_{-}|^{2}\approx\exp(-4\pi\tilde{\omega}_{f})\ll 1\), so that the adiabatic invariance of the magnetic moment is preserved. On the other hand, the same Equation (30)yields \(|u_{-}|^{2}\approx 1\) for \(\omega_{f}<0\) (provided \(|\hat{\omega}_{f}|\gg 1\)). We see that two different functions \(\omega(t)\), given by Equation (27) with \(k=1\) and Equation (29), yield identical _absolute values_ of the adiabatic coefficients \(|u_{\pm}|\). A common feature of these two functions is the _linear_ time dependence of the transition through zero value.  
However, the _phases_ of adiabatic coefficients are sensitive to the exact form of function \(\omega(t)\). For example, it was shown in [23] that function (29) with \(\omega_{f}=-\omega_{i}\) yields the complex adiabatic coefficients  
\[u_{+}=\sqrt{2}\,\exp[-4i\tilde{\omega}_{0}\ln(2)],\qquad u_{-}=i. \tag{31}\]  
We see that phases of coefficients \(u_{+}\) given by Equation (28) with \(k=1\) and Equation (31) are different.