Lower bounds on the growth of Sobolev norms in some linear time dependent Schr\"odinger equations

In this paper we consider linear, time dependent Schr\"odinger equations of the form $i \partial_t \psi = K_0 \psi + V(t) \psi $, where $K_0$ is a positive self-adjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant. We give a strategy to construct bounded perturbations $V(t)$ such that the Hamiltonian $K_0 + V(t)$ generates unbounded orbits. We apply our abstract construction to three cases: (i) the Harmonic oscillator on $\mathbb R$, (ii) the half-wave equation on $\mathbb T$ and (iii) the Dirac-Schr\"odinger equation on the sphere. In each case, $V(t)$ is a smooth and periodic in time pseudodifferential operator and the Schr\"odinger equation has solutions fulfilling $\| \psi(t) \|_r \gtrsim |t|^{r }$ as $|t| \gg 1$.