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Direct and Inverse Spectral Problems on Quantum GraphsWang, Tui-En 30 July 2012 (has links)
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.
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Ambarzumyan problem on treesLin, Chien-Ru 23 July 2008 (has links)
We study the Ambarzumyan problem for Sturm-Liouville operator defined on graph. The classical Ambarzumyan Theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator defined on
the interval [0,£k] are exactly {n^2: n ∈ N ⋃ {0} }, then the potential q=0. In 2005, Pivovarchik proved two similar theorems with uniform lengths a for the Sturm-Liouville operator defined on a 3-star graphs. Then Wu considered the Ambarzumyan problem for graphs
of nonuniform length in his thesis. In this thesis, we shall study the Ambarzumyan problem on more complicated trees, namely, 4-star graphs and caterpillar graphs with edges of different lengths. We
manage to solve the Ambarzumyan problem for both Neumann eigenvalues and Dirichlet eigenvalues. In particular, the whole spectrum can be partitioned into several parts. Each part forms the solution to one
Ambarzumyan problem. For example, for a 4-star graphs with edge lengths a, a, 2a, 2a form the solution to 3 different Ambarzumyan problems for the Neumann eigenvalues.
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