Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations

In this paper we consider linear, time dependent Schrodinger equations of the form i partial derivative(t)psi = K-0 psi + V(t)psi, where K-0 is a positive selfadjoint 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 Har- monic oscillator on N, (ii) the half-wave equation on and (iii) the Dirac-Schrodinger equation on Zoll manifolds. In each case, V(t) is a smooth and periodic in time pseudodifferential operator and the Schrodinger equation has solutions fulfilling the optimal lower bound estimate parallel to psi(t)parallel to r greater than or similar to vertical bar t vertical bar as vertical bar t vertical bar >> 1.