Lower bounds on the growth of Sobolev norms in some linear time dependent Schr\"odinger equations

TL;DR

This paper develops a method to construct bounded, time-periodic perturbations in linear Schrödinger equations that cause solutions to exhibit unbounded growth in Sobolev norms over time, demonstrating potential instability in such systems.

Contribution

It introduces a general strategy for creating perturbations leading to unbounded Sobolev norm growth in linear Schrödinger equations with discrete spectra.

Findings

Constructed explicit perturbations causing unbounded Sobolev norm growth.

Applied the method to harmonic oscillator, half-wave, and Dirac-Schrödinger equations.

Solutions grow as |t|^r in Sobolev norms for large |t|.

Abstract

In this paper we consider linear, time dependent Schr\"odinger 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-Schr\"odinger equation on the sphere. In each case, $V(t)$ is a smooth and periodic in time pseudodifferential operator and the Schr\"odinger equation has solutions fulfilling $\| \psi(t) \|_r \gtrsim |t|^{r }$ as $|t| \gg 1$.