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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 2139792, 14 pages
https://doi.org/10.1155/2018/2139792
Research Article

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

1School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2School of Mathematics and Information Engineering, Liupanshui Normal College, Liupanshui, Guizhou 553004, China

Correspondence should be addressed to Yangrong Li; nc.ude.uws@ryil

Received 27 September 2017; Accepted 19 December 2017; Published 17 January 2018

Academic Editor: Xiaohua Ding

Copyright © 2018 Lianbing She et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with pullback dynamics for the weakly damped Schrödinger equation with time-dependent forcing. An increasing, bounded, and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable. Also, we obtain the forward absorption, which is only used to deduce the backward compact-decay decomposition according to high and low frequencies. Based on a new existence theorem of a backward compact pullback attractor, we show that the nonautonomous Schrödinger equation has a pullback attractor which is compact in the past. The method of energy, high-low frequency decomposition, Sobolev embedding, and interpolation are quite involved in calculating a priori pullback or forward bound.