Spatial growth of fundamental solutions for certain perturbations of the harmonic oscillator

We consider the fundamental solution for the Cauchy problem for perturbations of the harmonic oscillator by time dependent potentials which grow at spatial infinity slower than quadratic but faster than linear functions and whose Hessian matrices have a fixed sign. We prove that the fundamental solution at resonant times grows indefinitely at spatial infinity with an algebraic growth rate, which increases indefinitely when the growth rate of perturbations at infinity decreases from the near quadratic to the near linear ones.