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

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 Zoll manifolds. In each case, $V (t)$ is a smooth and periodic in time pseudodifferential operator and the Schrödinger equation has solutions fulfilling the optimal lower bound estimate ${\lVert \psi (t) \rVert}_r \gtrsim {\lvert t \rvert}^r$ as ${\lvert t \rvert} \gg 1$.

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Received 2 February 2018

Accepted 12 July 2018

Published 25 October 2019