Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations
type: Item
instance of: scholarly article
title: Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations
zbMATH Open document ID: 1435.35139
author: Alberto Maspero
published in: Mathematical Research Letters
publication date: 2019-10-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1801.06813
review text: 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.
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 7124455
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: harmonic oscillator
zbMATH Keywords: half-wave equation
zbMATH Keywords: Dirac-Schrödinger equation
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
arXiv ID: 1801.06813
DOI: 10.4310/MRL.2019.V26.N4.A11
description: scientific article