  We consider Hamiltonian deformations of Gabor systems, where the window
evolves according to the action of a Schr\"odinger propagator and the
phase-space nodes evolve according to the corresponding Hamiltonian flow. We
prove the stability of the frame property for small times and Hamiltonians
consisting of a quadratic polynomial plus a potential in the Sj\"ostrand class
with bounded second order derivatives. This answers a question raised in [de
Gosson, M. Symplectic and Hamiltonian Deformations of Gabor Frames. Appl.
Comput. Harmon. Anal. Vol. 38 No.2, (2015) p.196--221.]
