  We have obtained explicit analytical formulas for the mean energy and its
variance (characterizing the energy fluctuations) of a quantum harmonic
oscillator with time-dependent frequency in the adiabatic regimes after the
frequency passes through zero. The behavior of energy turns out to be quite
different in two cases: when the frequency remains real and when it becomes
imaginary. In the first case, the mean energy always increases when the
frequency returns to its initial value, and the increment coefficient is
determined by the exponent in the power law of the frequency crossing zero. On
the other hand, if the frequency becomes imaginary, the absolute value of mean
energy increases exponentially, even in the adiabatic regime, unless the
Hamiltonian becomes time independent. Small corrections to the leading terms of
simple adiabatic approximate formulas are crucial in this case, due to the
unstable nature of the motion.
