Gaussian beam methods for the Schrӧdinger equation with periodic potentials and strictly hyperbolic systems

Abstract

I would like to take this opportunity to express my gratitude to those who helped me with conducting my research and writing this thesis. First of all, my advisor Prof. Hailiang Liu for his guidance, patience, support and keeping me motivated to work hard. I learned a lot from him during my graduate study and this knoweledge will help me in the future work. I appreciate his time and energy he devoted for my training during these years. Also I want to thank my coleagues and classmates, in particular, our members of the Focused Research Interaction seminar organized by Prof. Liu: Dr. Hui Yu, Nattapol Ploymaklam and Yongki Lee; presenting our research and communicating with each other helped me to understand my research topic better. I appreciate all suggestions and comments from my committee members, in particular Prof. Paul Sacks for his careful review and also for his deep questions which would be interesting to address in the future. I would like to thank all my ommittee members Prof. G. Lieberman, Prof. A. Roiterstein and Prof. M. Smiley. I feel very grateful to the Department of Mathematics of Iowa State University for providing me this life changing opportunity for graduate study and work. Finally, I would like to thank the members of "SQuaRE" research group, which is working on Gaussian beam methods. In particular,