  In this work we study the persistence in time of superoscillations for the
Schr\"{o}dinger equation with quadratic time-dependent Hamiltonians. We have
solved explicitly the Cauchy initial value problem with three different kind of
oscillatory initial data. In order to prove the persistence of
superoscillations we have defined explicitly an operator in terms of solutions
of a Riccati system associated with the variable coefficients of the
Hamiltonian. The operator is defined on a space of entire functions. Particular
examples include Caldirola-Kanai and degenerate parametric harmonic oscillator
Hamiltonians and more. For these examples we have illustrated numerically the
convergence on real and imaginary parts.
