Inverse Problems For Two Kinds Of Differential Operators On Graphs

Differential operators on graphs,also called quantum graphs,arise in quantum mechanics,nanotechnology,organic chemistry,electronics and so on.For example,the motion of electrons in carbon nano-structures can be described by quantum graphs.So far a few results have been obtained for differential operators on graphs,most of which consider direct problems.However,the related inverse problems are still not sufficiently investigated.In this thesis,we study two kinds of differential operators on graphs—Sturm-Liouville operators and Dirac operators—and obtain some uniqueness theorems and reconstruction algorithms of the related inverse problems,which consist in recovering unknown potentials from observable quantities.The results in this thesis give the well-posedness of the solutions of the related inverse problems and enrich the spectral theory of differential operators.In addition,they can be applied to related problems in quantum mechanics and other fields.The main research contents of this thesis are as follows.In the first chapter,we introduce some application backgrounds of quantum graphs and the history of researches on such operators.We also summarize the main work and new ideas of this thesis.In the second chapter,we are concerned with partial inverse problems of Sturm-Liouville operators and Dirac operators on a star graph.For each operator,it is shown that if the potentials are known on all but one edges,then several eigenvalue sequences and a part of unknown potential can uniquely determine the whole potential.In the third chapter,we investigate the inverse vertex observation problem for the heat equation on a star graph with three edges.Firstly,we classify the eigenvalues of the SturmLiouville operator on the star graph.Then applying this classification,the inverse Laplace transform,Kramer’s sampling theorem and the Gelfand-Levitan theory,it is proven that the potentials are uniquely determined by the observations on the vertices,and the reconstruction algorithm is also given.In the fourth chapter,we study weight matrices of the Dirac operator on a star graph and the related inverse spectral problems.Firstly,we give the definitions of Weyl function and weight matrices of the operator.Secondly,we prove the nonnegative definiteness of the weight matrices by matrix calculations and derive the asymptotics of the weight matrices by the method of contour integral.Then a Riesz basis consisting of eigenfunctions is constructed according to these results.On the other hand,we prove the uniqueness theorem of recovering the potentials from the Weyl function and obtain the algorithm of reconstructing the potentials from the spectrum and the sequence of weight matrices of the operator.