Direct and Inverse Spectral Problems on Quantum Graphs

• Main Authors: Tui-En Wang, 王推恩
• Other Authors: Chun-Kong Law
• Format: Others
• Language: en_US
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/41613722815589884488

博士 === 國立中山大學 === 應用數學系研究所 === 100 === Recently there is a lot of interest in the study of Sturm-Liouville problems on graphs, called quantum graphs. However the study on cyclic quantum graphs are scarce. In this thesis, we shall rst consider a characteristic function approach to the spectral analysis for the Schrodinger operator H acting on graphene-like graphs|in nite periodic hexagonal graphs with 3 distinct adjacent edges and 3 distinct potentials de ned on them. We apply the Floquet-Bloch theory to derive a Floquet equation with parameters theta_1, theta_2, whose roots de ne all the spectral values of H. Then we show that the spectrum of this operator is continuous. Our results generalize those of Kuchment-Post and Korotyaev-Lobanov. Our method is also simpler and more direct. Next we solve two Ambarzumyan problems, one for graphene and another for a cyclic graph with two vertices and 3 edges. Finally we solve an Hochstadt-Lieberman type inverse spectral problem for the same cyclic graph with two vertices and 3 edges. Keywords : quantum graphs, graphene, spectrum, Ambarzumyan problem, inverse spectral problem.