In a very elementary exercise the authors consider the superposition
of $P^2$ and $Q^2$ with the respective time-dependent coefficients
$\alpha^2/2$ and $\beta^2/2$ and call it the time-dependent
quadratic Hamiltonian (TDQH). The same operator may be re-expressed
in terms of the usual time-independent annihilation and creation
operators $a$ and $a^\dagger$ of course. By the time-dependent
harmonic oscillator (TDHO) the authors mean again the same
Hamiltonian operator expressed in terms of the (ad hoc, tilded)
time-dependent creation and annihilation operators. The other set
$b$ and $b^\dagger$ of the time-dependent annihilation and creation
operators is finally introduced via Bogoliubov transformation
mediated by the Wei-Norman system-evolution operator $U(t)$
expressed as a product of three exponentials. Its three parameters
$g_j$ with $j = 1,2,3$ are finally expressed in terms of $\alpha$
and $\beta$. Two well known solvable examples are recalled for
illustration.


MR2718324 Urdaneta, Inés; Sandoval, Lourdes; Palma, Alejandro On the
algebraic approach to the time-dependent quadratic Hamiltonian. J.
Phys. A 43 (2010), no. 38, 385204, 10 pp. 81Qxx