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

Abstract

In this paper we consider linear, time dependent Schrödinger 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-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 $\Vert \psi (t){\Vert }_r\gtrsim |t{|}^r$ as $|t|\gg 1$.

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