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

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In this paper we consider linear, time dependent Schrödinger equations of the form i ∂ₜ ψ= K₀ ψ+ V(t) ψ, where K₀ 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 K₀ + V(t) generates unbounded orbits. We apply our abstract construction to three cases: (i) the Harmonic oscillator on R, (ii) the half-wave equation on T and (iii) the Dirac-Schrödinger equation on the sphere. In each case, V(t) is a smooth and periodic in time pseudodifferential operator and the Schrödinger equation has solutions fulfilling ψ(t) ᵣ |t|ʳ as |t| ≫ 1.