Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations
This paper deals with the following linear Schrödinger equation: \[i\partial_t\psi=K_0\psi+V(t)\psi\tag{1}\] on a class of Hilbert spaces \(\mathcal{H}^r\), where \(K_0\) is a positive, selfadjoint operator with purely discrete spectrum and whose spectral gaps are asymptotically constant, \(V(t)\) is a time depenent self-adjoint perturbation and the scale \(\mathcal{H}^r\) is defined spectrally by \(K_0\). The author develops a technique to construct bounded operators \(V(t)\), which are smooth and periodic in \(t\) pseudodifferential operators, for which (1) possesses unbounded orbits. It is shown that the norms of the solutions fulfill the optimal lower bound estimate \(\|\psi(t)\|_r\gtrsim|t|^r\) as \(|t|\gg 1\), despite every orbit of the unperturbed flow is bounded. Applications are given to the harmonic oscillator on \(\mathbb{R}\), the half-wave equation and to the Dirac-Schrödinger equation on Zoll manifolds.
scientific article