Abstract |
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,π] 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. |